Certain Classes of Codes from Combinatorial Designs · 12 J. Yin, Sept. 14, 2006 Japan Seminar...

Certain Classes of Codes fromCombinatorial Designs

Yin Jianxing

Department of Mathematics,

Suzhou University

September 14, 2006

J. Yin, Sept. 14, 20061

Japan Seminar Codes From Combinatorial Designs

1.1 Constant Composition Codes

• An (n,M, d; q)-code is a code C ⊆ Qn with length n, size

M(= |C|) and Hamming distance d(C) = d.

• Here, Q is an alphabet of q elements called symbols or letters.

Most of the time Q is taken to be the finite field GF(q) of order q

or the ring of integers modulo q.

• An (n,M, d; q)-code is refereed to as a constant weight code

(CWC), or an (n,M, d, w; q)-CWC, if its codewords have the

same weight w.

J. Yin, Sept. 14, 20062


• Let Q = {at : 0 ≤ t ≤ q − 1}. An (n,M, d; q)-code over Q is

refereed to as a constant composition code (CCC), or an

(n,M, d, [w0, w1, · · · , wq−1]; q)-CCC, if for any i (0 ≤ i ≤ q − 1),

the symbol ai appears exactly wi times in every codeword.

• The constant composition [w0, w1, · · · , wq−1] is called the type

of the CCC, which is essentially an unordered multiset. We will

write it in an exponential notation in the sequel. In case the wi

are themselves exponents, we revert to the composition list to

avoid confusion.

J. Yin, Sept. 14, 20063


• CCCs arise in frequency hopping, when a schedule is needed to

determine frequencies on which to transmit. When each frequency

is to be used a specified number of times within a frame, each

frequency hopping sequence is a codeword of constant compo-

sition. Indeed whenever a different cost is associated with each

symbol in the underlying alphabet, uniform cost of codewords

leads to constant composition specified number of time, see Chu,

Colbourn and Dukes (DAM 2006).

• CCCs are also useful in the powerline communication.

J. Yin, Sept. 14, 20064


• Constant composition codes are also a subclass of constant

weight codes by definition.

• The class of binary constant composition codes coincides with the

class of binary constant weight codes

• An (n,M, d, [w0, w1, . . . , wq−1]; q)-CCC is called a permutation

code if n = q and wi = 1 for all i. Hence, permutation codes are

a special class of constant composition codes.

J. Yin, Sept. 14, 20065


1.2 Bounds on CCCs

• One of the most fundamental problems in combinatorial

coding theory is to determine the maximum size of a

block code when its other parameters have been fixed.

• The notation A(n, d, [w0, w1, ..., wq−1]; q) stands for the maximum

size of an (n,M, d, [w0, w1, ..., wq−1]; q) constant composition code.

A CCC with A(n, d, [w0, w1, ..., wq−1]; q) codewords is said to be

optimal.

J. Yin, Sept. 14, 20066


• To measure the optimality of CCCs, there are two classic upper

bounds that we can employ: the Johnson and the Plotkin bounds.

The latter holds for any code.

• Plotkin Bound: If an (n,M, d; q)-code with d > n(q − 1)/q

exists, then

M ≤ qd

qd− n(q − 1).

J. Yin, Sept. 14, 20067


• The Johnson bounding technique (Johnson (1962)) was

applied to ternary CCCs by Svanstrom et al. (IEEE IT 2002):

• Johnson Bound: For any integer r satisfying 0 ≤ r ≤ q − 1,

A(n, d, [w0, w1, · · · , wq−1]; q) ≤ n

wr

A(n− 1, d, [w0, w1, · · · , wq−1]; q)

where

wi =

{wi − 1, if i = r

wi, if i 6= r.

J. Yin, Sept. 14, 20068


• Luo, Fu, Han Vink and Chen (IEEE IT 2003) established the

following bound for CCCs.

LFVC Bound: If nd− n2 + (w20 + w2

1 + · · ·+ w2q−1) > 0, then

A(n, d, [w0, w1, ..., wq−1]; q) ≤ nd

nd− n2 + (w20 + w2

1 + · · ·+ w2q−1)

.

J. Yin, Sept. 14, 20069


• (Ding and Yin (IEEE IT 2005-2)) Let

N =(w2

0 + w21 + · · ·+ w2

q−1

)− λn;

λ = gcd {wi| i = 0, 1, · · · , q − 1};N = Nλ.

If N > 0, n(n− λ) ≡ 0 (mod N) and n− λ 6≡ 0 (mod N), then

A(n, n− λ, [w0, w1, · · · , wq−1]) ≤ n(n− λ)

N− 1.

J. Yin, Sept. 14, 200610


• Combining LFVC Bound with the Johnson Bound produces

the following.

Lemma 1.1 For any (n,M, n− λ, [λ, λ, ..., λ]; q)-CCC,

M ≤ q(n− 1),

where n = λq.

J. Yin, Sept. 14, 200611


1.3 The Comb. Characterization

• Suppose that there is a set X of v points and that from these

a collection A of subsets, or blocks, is drawn. The ordered pair

(X,A) is called a design of order v.

• In design theory there are normally a number of additional rules

imposed when the blocks are selected.

• A design is called an (n, λ)-packing if there exists a (minimum)

constant λ such that every pair of distinct points occurs in at

most λ blocks, and every point occurs in precisely n blocks.

J. Yin, Sept. 14, 200612


• An α-parallel class of a design is a set of blocks such that each

point occurs in precisely its α blocks. When α = 1, it is simply

termed a parallel class.

• A resolution of the design is a partition of its blocks into

α-parallel classes for certain values of α.

• Two resolutions of a design are said to be orthogonal if any class

in one resolution intersects every class from the other resolution in

at most one block. Here, repeated blocks (if they exist) are

regarded as distinct blocks.

J. Yin, Sept. 14, 200613


• A generalized doubly resolvable packing (GDRP), or a

GDRP(n, λ; v) of type {λ0, λ1, · · · , λm−1} is an (n, λ)-packing of

order v whose blocks can be arranged into an m× n array R with

the following properties.

i. Each cell of R is either empty or contains one block.

ii. For 0 ≤ i ≤ m− 1, the blocks in row i of R form a λi-parallel

class.

iii. The blocks in every column of R form a parallel class.

J. Yin, Sept. 14, 200614


• The type {λ0, λ1, · · · , λm−1} is described by an exponential

notation as with CCCs.

• The definition implies n =∑m−1

i=0 λi. Hence, whenever the

exponential notation ga11 ga2

2 · · · gass is used, we have

n = a1g1 + a2g2 + · · ·+ asgs and m = a1 + a2 + · · ·+ as.

J. Yin, Sept. 14, 200615


• CCCs can be characterized by GDRPs:

Theorem 1.2 (Ding and Yin (IEEE IT 2005-2))

The existence of a GDRP(n, λ; v) of type {λ0, λ1, · · · , λq−1} is

equivalent to that of an (n,M, d, [w0, w1, · · · , wq−1]; q)-CCC,

where

M = v, d = n− λ and λj = wj, 0 ≤ j ≤ q − 1.

J. Yin, Sept. 14, 200616


Example:

A (10, 10, 8, [4123]; 4)−CCC

0 3 2 3 1 0 2 1 0 0 0

1 0 3 2 3 1 0 2 1 0 0

2 0 0 3 2 3 1 0 2 1 0

3 0 0 0 3 2 3 1 0 2 1

4 1 0 0 0 3 2 3 1 0 2

5 2 1 0 0 0 3 2 3 1 0

6 0 2 1 0 0 0 3 2 3 1

7 1 0 2 1 0 0 0 3 2 3

8 3 1 0 2 1 0 0 0 3 2

9 2 3 1 0 2 1 0 0 0 3

⇐⇒

A GDRP(10, 2; 10) of type 4123

(transposed)

0 1 2 3

f1,2,3,6g f4,7g f5,9g f0,8gf2,3,4,7g f5,8g f0,6g f1,9gf3,4,5,8g f6,9g f1,7g f0,2gf4,5,6,9g f0,7g f2,8g f1,3gf0,5,6,7g f1,8g f3,9g f2,4gf1,6,7,8g f2,9g f0,4g f3,5gf2,7,8,9g f0,3g f1,5g f4,6gf0,3,8,9g f1,4g f2,6g f5,7gf0,1,4,9g f2,5g f3,7g f6,8gf0,1,2,5g f3,6g f4,8g f7,9g

J. Yin, Sept. 14, 200617


1.3 CCCs from Frequency Rectangles

• A frequency rectangle of type FR(m,λ) is an m× (λm)

matrix over an m-set S such that every element of S occurs

exactly λ times in each row and once in each column.

• The positive integer λ is called the frequency of elements in the

rectangle. When λ = 1, it is nothing else than Latin square of

order m.

J. Yin, Sept. 14, 200618


• Two frequency rectangles F1 and F2 of type FR(m,λ) over S are

said to be orthogonal if, when F2 is superimposed on F1, every

ordered pair in S × S appears precisely λ times.

• A set of λm− 1 mutually orthogonal frequency rectangles (or

simply MOFRs) of type FR(m,λ) is called complete. When

λ = 1, it is a complete set of m− 1 mutually orthogonal latin

squares of order m.

J. Yin, Sept. 14, 200619


Theorem 1.3 If there exists a complete set of λq − 1 MOFRs of type

FR(q, λ), then so does an optimal

(λq, q(λq − 1), λ(q − 1), [λq]; q)-CCC,

whose size meets the bound in Lemma 1.1.

J. Yin, Sept. 14, 200620


Theorem 1.4 For any prime power q and positive t, there exists a

complete set of qt − 1 MOFRs of type FR(q, qt−1) over GF(q), or

equivalently an optimal

(qt, q(qt − 1), qt − qt−1, [qt−1, qt−1, · · · , qt−1]; q)-CCC.

J. Yin, Sept. 14, 200621


• The Proof of Theorem 1.4

Consider the qt − 1 distinct polynomials over GF(q) of the form

fα, β1,...,βt(x, y1, . . . , yt) = αx + β1y1 + . . . + βtyt,

where α 6= 0, (β1, · · · , βt) 6= (0, . . . , 0) and no two of them are

nonzero multiples of each other.

J. Yin, Sept. 14, 200622


For any such polynomial, we construct a q × qt matrix F overGF(q) so that its rows are indexed by q elements of GF(q) andcolumns are indexed by qt t-tuples over GF(q), and the entry in thecell (a, (b1, . . . , bt)) is fα, β1,··· ,βt(a, b1, . . . , bt). This produces acomplete set of qt − 1 MOFRs of type FR(q, qt−1) over GF(q)

J. Yin, Sept. 14, 200623


1.4 CCCs from Nonlinear Functions

• We use Carlet and Ding (J. Complexity 2004), Coulter

and Matthews (DCC 1997) as our key references on

nonlinear Functions.

• Let (A, +) and (B, +) be two finite abelian group. A function

f : A −→ B is called linear if f(x + y) = f(x) + f(y) for all

x, y ∈ A.

J. Yin, Sept. 14, 200624


• A robust measure of the nonlinearity of a function f : A −→ B

using the derivatives Daf(x) = f(x + a)− f(x) is given by

Pf = max06=a∈A

maxb∈B

|{x ∈ A : Daf(x) = b}||A| ,

where |A| denotes the cardinality of the set A.

J. Yin, Sept. 14, 200625


• For any fixed element a ∈ A, the sets {x ∈ A : Daf(x) = b}(b ∈ B) constitute a partition of A, and hence

|A| =∑

b∈B

|{x ∈ A : Daf(x) = b}| ,

namely,

∑

b∈B

|{x ∈ A : Daf(x) = b}||A| = 1.

J. Yin, Sept. 14, 200626


• The maximum value of the |B| non-negative integers

|{x ∈ A : f(x + a)− f(x) = b}||A| , b ∈ B

is greater than or equal to its mean for any a ∈ A \ {0}. Therefore,

Pf ≥ 1

|B| .

J. Yin, Sept. 14, 200627


• The smaller the value of Pf , the higher the corresponding

nonlinearity of f . If f is linear, then Pf = 1.

• A function f : A −→ B is called a perfect nonlinear function

(PNF) if it has perfect nonlinearity, that is, Pf = 1|B| .

J. Yin, Sept. 14, 200628


• Since the maximum of a sequence of numbers equals its mean if

and only if the sequence is constant, Pf = 1|B| if and only if, for

every b ∈ B and every a ∈ A∗ = A \ {0}, the quantity

|{x ∈ A : Daf(x) = b}| has value |A||B| (this is possible only if

|B| divides |A|).• A perfect nonlinear function f : GF(2)n −→ GF(2) (n > 1) is a

bent function.

J. Yin, Sept. 14, 200629


• A perfect nonlinear function from a finite abelian group to a finite

abelian group of the same order is called a planar function in

finite geometry.

• Planar functions were introduced by Dembowski and Ostrom

(1968) for the construction of affine planes.

• Perfect nonlinear functions introduced by Nyberg (1992) and bent

functions introduced by Rothaus (1976) are extensions of planar

functions.

J. Yin, Sept. 14, 200630


The Encoding Method

Let Π be a PFN from an abelian group (A, +) of order n to an

abelian group (B, +) of order m. Write

A = {a0, a1, · · · , an−1}, B = {b0, b1, · · · , bm−1}.Define

wi = |{x ∈ A : Π(x) = bi}|for each i, and

CΠ = {(Π(a0 + ai), · · · , Π(an−1 + ai)) : 0 ≤ i ≤ n− 1}.

J. Yin, Sept. 14, 200631


• Theorem 1.5 (Ding and Yin (IEE IT 2005-1)) The CΠ defined

above is an (n, n, (m− 1)n/m, [w0, w1, · · · , wm−1; m]) CCC, and is

optimal with respect to the LFVC Bound.

• Applying Theorem 1.5, the known perfect nonlinear functions can

be employed to obtain optimal CCCs, provided that we are able to

determine the frequencies wi. Below is one sample.

J. Yin, Sept. 14, 200632


Example: Let Π(x) =∑t

i=1 x2i−1x2i be the PNF from GF(q)2t to

GF(q). Consider the equation

t∑i=1

z2i−1z2i = a, a ∈ GF(q).

It has Ca solutions of (z1, z2, · · · , z2t) ∈ GF(q)2t. It can be proved that

Ca =

{q2t−1 + (q − 1)qt−1, a = 0

q2t−1 − qt−1, a 6= 0.

J. Yin, Sept. 14, 200633


• It follows that by Theorem 1.5 CΠ defined above is a

(q2t, q2t, (q − 1)q2t−1,

[q2t−1 + (q − 1)qt−1, q2t−1 − qt−1, · · · , q2t−1 − qt−1])

CCC, and is optimal with respect to the LFVC Bound.

J. Yin, Sept. 14, 200634


1.5 CCCs from Partitioned DFs

• Let F = {D1, D2, · · · , Ds} be a collection of subsets (called base

blocks) of an additive abelian group G of order v, and

K = [|D| : D ∈ F ], denoted by an exponential form, is the list of

sizes of base blocks. If the difference list (multiset)

∆F =s⋃

i=1

∆Di =s⋃

i=1

{a− b : a, b ∈ Di and a 6= b}

contains every nonzero element exactly λ times, then F is said to

be a (v, K, λ) difference family, or a (v, K, λ)-DF for short, in G.

J. Yin, Sept. 14, 200635


• The above definition is to say that F is a (v, K, λ)-DF (or PDF)

iff the function

dF(g) :=∑D∈F

|(D + g) ∩D|

takes on a constant value λ when w ranges over all the nonzero

elements of G, where D + g = {x + g : x ∈ D}.• This function is referred to as a difference function defined on

G \ {0}.

J. Yin, Sept. 14, 200636


• In the case that F form a partition of G, we call it partitioned,

or a (v, K, λ)-PDF.

• Theorem 1.6 (Ding and Yin (IEE IT 2005-2))

If a (v, [λ0, λ1, . . . , λq−1], λ)-PDF exists, then so does an optimal

GDRP(n, λ; v) of type {λ0, λ1, · · · , λq−1}, or equivalently an

optimal (n, v, n− λ, [λ0, λ1, . . . , λq−1]; q) CCC (meeting the LFVC

Bound).

J. Yin, Sept. 14, 200637


2.1. Deletion-Correcting Codes

• The conventional error-correcting codes deal with the errors of

symmetric form

x = abccdef −→ y = afcedeb.

Here, Hamming distance d(x,y) = 3.

• During transmission of information strings there exist the errors of

the asymmetric form

abccdef −→ abcdf (2 symbols are deleted)

abccdef −→ abcacdecf (2 symbols are added)

J. Yin, Sept. 14, 200638


• Whenever asymmetric errors occur, the length of the received

string is generally different from the length of the transmitted

string. This makes it impossible to use Hamming metric as a

measure for the distance between two sequences.

• Let x ∈ Qn and y ∈ Qm, where Q stands for an alphabet of size q.

The Levenshtein distance l(x,y) between x and y is defined to

be the smallest number of deletions and insertions needed to

change x to y.

J. Yin, Sept. 14, 200639


• Let Dt(c) denote the set of vectors a ∈ Qn−t that are obtained by

deleting any t components from c. A code C ⊆ Qn is said to be a

t-deletion-correcting code if Dt(c1)⋂

Dt(c2) = ∅ for all

c1, c2 ∈ C with c1 6= c2.

• For any code C ⊆ Qn, define

l(C) = min{l(x,y) : x,y ∈ Qn and x 6= y}.A code C is a t-deletion-correcting code if and only if l(C) > 2t.

J. Yin, Sept. 14, 200640


• A code c is capable of correcting t insertion and/or deletions if

and only if l(C) > 2t. Hence, a t-deletion-correcting code can

correct any combination of up to t deletions and insertions.

• A t-deletion-correcting code code C ⊆ Qn is called perfect, or a

T ∗(n− t, n, q)-code, if Dt(c) (c ∈ C) partition Qn−t.

J. Yin, Sept. 14, 200641


• The possibility of packet loss on internet transmissions has

renewed interest in deletion-correcting codes. The problem is this:

you send n symbols, but only n− t arrive. The t symbol are

deleted, but neither you nor the receiver know which of the n

symbols were lost. One needs to design a code that can correct

such errors.

• D’Yachkov et al. (J. Combin. Optimization 2003) developed a link

between deletion-correcting codes and DNA-Codes.

J. Yin, Sept. 14, 200642


• To my knowledge, very little is known about the theory of

deletion-correcting codes.

• Research on deletion-correcting codes in combinatorial community

has mainly concentrated on the existence problem of a

T ∗(n− t, n, q)-code.

J. Yin, Sept. 14, 200643


The known T ∗(n− t, n, q)-codes

¨ For any positive integer v, a T ∗(2, k, v)-code exists whenever

3 ≤ k ≤ 5 (Levenshtein (DMA 1992); Bourse (DCC 1995);

Mahmoodi (DCC 1998)), where the T ∗(2, 3, v)-codes obtained by

Levenshtein are largest.

¨ For any positive integer v, a T ∗(2, 6, v)-code exists with possible

exceptions of v ∈ {173, 178, 203, 208}(Yin (DCC 2001); Shalaby,

Wang and Yin (DCC 2002)).

J. Yin, Sept. 14, 200644


¨ (Levenshtein (DMA 1992) showed that a T ∗(3, 4, v) -code exists

for all even integers v and left the problem for odd integers open.

Recently, Wang and Ji (JCD 2005) tackled this problem

completely.

¨ A T ∗(2, 7, v)-code exists with a handle of possible exceptions of v

(Wang and Yin (IEEE IT 2006).

J. Yin, Sept. 14, 200645


• The another problem is to find the largest codes. As before, let

A(n, t; q) denote the maximum size of a t-deletion-correcting code

C ⊆ Qn. Such a code with A(n, t; q) codewords is said to be

optimal.

• It is much more difficult generally to obtain upper bounds for

deletion-correcting codes than for conventional error-correcting

codes, since the disjoint code spheres Dt(c) associated with the

codewords do not all have the same size.

J. Yin, Sept. 14, 200646


• The upper bound in the size of a T ∗(2, k, v)-code was given by

Bours (DCC 2005) which we mention later.

• Levenshtein showed that A(n, 1; 2) ∼ 2n

n, as n →∞.

• Besides, very little is known about the values of A(n, t; q). Even

for q = 2 and t = 1, the value of A(n, 1; 2) with n ≥ 9 hasn’t yet

determined.

J. Yin, Sept. 14, 200647


The equivalent definition of T ∗(t, k, v)-codes

• A word (vector) x ∈ Qm is said to be a subword of a word

x ∈ Qn (m ≤ n) if x can be obtained from y by deleting (n−m)

symbols, or equivalently y can be obtained from x by inserting

(n−m) symbols.

• A perfect (k − t)-deletion-correcting code, or a T ∗(t, k, v)-code,

is a subset C ⊆ Qk such that every word of Qt occurs as a

subword in exactly one codeword of C. Here Q is the alphabet of

cardinality v.

J. Yin, Sept. 14, 200648


• From this definition, we see easily that a T ∗(t, k, v)-codes is

capable of correcting any combination of up to (k − t) deletions

and insertions of symbols, since any two distinct codewords in C

cannot share a common subword of length t or longer.

• This definition also suggests a link between T ∗(t, k, v)-codes and

transitive Stainer systems in design theory. If C is a T ∗(2, k, v)-

code, we can verify readily that

|C| ≤v

⌊2(v−1))

k−1

⌋

k

+ v

J. Yin, Sept. 14, 200649


T ∗(2, k, v)-codes from directed BIBD

• A set of k elements {a1, a2, · · · , ak} is said to be transitively

ordered by ai < aj for 1 ≤ i < j ≤ k. In other words, a

transitively ordered k-tuple (a1, a2, · · · , ak) consists of (k − 1)k/2

ordered pairs (ai, aj), 1 ≤ i < j ≤ k.

• A directed (DBIBD) with block size k and order v, denoted by

DB(k, 1; v), is a design (X,A) where A is a set of transitively

ordered k-tuples (blocks) of X, such that every ordered pair of

distinct points of X occurs in exactly one block.

J. Yin, Sept. 14, 200650


• Given a DB(k, 1; v), according to its definition, we may take its

point set as the alphabet Q. Then its blocks form

2v(v − 1)/k(k − 1) codewords of length k, in which every pair of

distinct symbols of Q occurs as a “subword” exactly once, while

any pair of the form (x, x) does not occur. It follows that a

T ∗(2, k, v)-code over Q can be obtained by taking all blocks of the

DB(k, 1; v) and all sequences of length k of the form (x, x, · · · , x),

where x runs over all symbols of Q. Furthermore, we have the

following construction.

J. Yin, Sept. 14, 200651


• Theorem 2.1 (Bours (DCC 1995)) Suppose that k ≥ 3 is an

integer and a DB(k, 1; v) exists. Then a T ∗(2, k, v + 1)-code and a

T ∗(2, k, v − e)-code exist, where 0 ≤ e ≤ 2 if k ≥ 5 and 0 ≤ e ≤ 3

if k ≥ 7.

J. Yin, Sept. 14, 200652


T ∗(2, k, v)-codes from IDBIBDs

• The use of a DB(k, 1; v) in Theorem 5.1 is restricted by its

parameters k and v, which must satisfy 2(v − 1) ≡ 0 (mod k − 1)

and 2v(v − 1) ≡ 0 (mod k(k − 1)). A more general construction

was established using an incomplete DBIBDs (briefly IDBIBDs).

• An IDBIBD, denoted by IDB(k, 1; v, w), is a DB(k, 1; v) (X,A)

missing the blocks of a DB(k, 1; w) (X,B), where Y ⊆ X and

B ⊆ A, and the subdesign, a DB(k, 1; w), is not necessarily to

exist.

J. Yin, Sept. 14, 200653


• Theorem 2.2 (Yin (DCC 2001)) Suppose that k ≥ 3 and an

IDB(k, 1; v, w) exists. Then

i. there exists a T ∗(2, k, v + 1)-code if a T ∗(2, k, w + 1)-code exists;

ii. there exists a T ∗(2, k, v − e)-code if a T ∗(2, k, w − e)-code exists

and 0 ≤ e ≤ min{w, bk−12c}.

• The resulting code in Theorem 2.2 contains a subcode, which is

quite useful in establishing some more combinatorial constructions

for T ∗(2, k, v)-codes.

J. Yin, Sept. 14, 200654


• Note that for the t ≥ 3 case, both the construction of a T ∗(t, k, v)

-code and the determination of the upper bound for its size are

very difficult task. This can be seen from the fact that a perfect

1-deletion-correcting code can be a T ∗(k − 1, k, v)-code.

J. Yin, Sept. 14, 200655


3.1 Signal Sets (Codebooks)

• An (N, K) signal set (or codebook) is a set C = {c0, ..., cN−1} of

N unit norm 1×K complex vector ci, which is called codewords

of the signal set.

• The alphabet of the signal set is the set of different complex values

that the coordinates of all the codewords take.

• The alphabet size is the number of elements in the the alphabet.

• Here and below the codewords are required to be normalized, i.e.,

the norm of every codeword must be 1.

J. Yin, Sept. 14, 200656


• The root-mean-square (RMS) correlation and the maximum

correlation amplitudes of such a signal set are defined as

Irms(C) :=

√√√√ 1

N(N − 1)

∑0≤i,j≤N−1

i 6=j

|〈ci, cj〉|

=

√√√√ 1

N(N − 1)

∑0≤i,j≤N−1

i 6=j

|cicHj |,

Imax(C) := max0≤i<j≤N−1

|cicHj |,

where cHj is the Hermite transpose of the 1×K complex vector cj,

and 〈ci, cj〉 = cicHj is the standard inner product.

J. Yin, Sept. 14, 200657


• Signal sets with best correlation property are desirable in

code-division multiple-access systems (CDMA). In the CDMA

application, a signal set is used to distinguish between the signals

of different users. To this end, the maximum correlation

amplitude Imax of the signal set should be as small as possible.

J. Yin, Sept. 14, 200658


• Direct sequence code division multiple access (DS-CDMA) is one

of the approaches to spread spectrum for signal transmission.

Optimal and almost optimal (N,K) signal sets with N > K are

desirable in synchronous DS-CDMA systems for reducing

interference, where the number of users N is greater than the

signal space dimension or the spreading factor K. Such signal sets

could be quite useful in 3G and 4G networks for increasing the

subscriber capacity within the limited spectral resource.

J. Yin, Sept. 14, 200659


• Welch’s bounds (Welch (IEEE IT 1974)) For any (N, K)

signal set C with N ≥ K,

Irms(C) ≥√

N −K

(N − 1)K, (1)

with equality if and only if∑N

i=0 cHi ci = (N/K)IK , where IK

denotes the K ×K identity matrix.

J. Yin, Sept. 14, 200660


• We have also

Imax(C) ≥√

N −K

(N − 1)K, (2)

with equality if and only if for all pairs (i, j) with i 6= j

|cicHj | =

√N −K

(N − 1)K. (3)

J. Yin, Sept. 14, 200661


• If the equality holds in (1), then C is referred to as a

Welch-bound-equality (WBE) signal set.

• A signal set meeting the bound of (2) is called a

maximum-Welch-bound-equality (MWBE) signal set.

• An MWBE signal set must be a WBE signal set, but a WBE

signal set may not be an MWBE signal set. MWBE signal sets

form a subset of WBE signal sets.

J. Yin, Sept. 14, 200662


• It is fairly easy to construct WBE signal sets. Every linear error

correcting code whose dual code with Hamming distance at least 3

yields a WBE signal set (Massey and Mittelholzer (1993); Sarwate

(1998)).

• However, MWBE signal sets are very hard to construct, as

pointed out by Sarwate (1998).

J. Yin, Sept. 14, 200663


• The known classes of MWBE signal sets:

¨ The trivial (N, N) and (N, N − 1) MWBE signal sets;

¨ The (N, K) MWBE signal sets based on conference matrices

(Conwayet al. (1996); Strohmer and Heath (2003)), when

N = 2K = 2d+1 and N = 2K = pd + 1 with p a prime number,

where d is a positive integer.

¨ (N, K) MWBE signal sets based on (N, K, λ)-DSs which we will

present later.

J. Yin, Sept. 14, 200664


• It was shown in (Strohmer and Heath (2003)) that

♦ No (N,K) real signal set C can meet the Welch bound of (2), if

N > K(K + 1)/2.

♦ No (N,K) signal set C can meet the Welch bound of (2), if

N > K2.

• From the above fact, we see that Welch’s bound on the maximum

correlation amplitude cannot be achieved in certain cases either,

as it is not tight in these cases.

J. Yin, Sept. 14, 200665


• The following bounds developed by Levenstein (Kabatyanskii and

Levenstein (1978); Levenstein (1983)) are better than Welch’s

bound in certain cases.

¨ For any (N,K) real signal set C with N > K(K + 1)/2, we have

Imax(C) ≥√

3N −K2 − 2K

(K + 2)(N −K). (4)

¨ For any (N, K) complex signal set C with N > K, we have

Imax(C) ≥√

2N −K2 −K

(K + 1)(N −K). (5)

J. Yin, Sept. 14, 200666


• Another lower bound documented in (Xia et al. (IEEE IT 2005) is

the following:

Imax(C) ≥ 1− 2N− 1K−1 . (6)

J. Yin, Sept. 14, 200667


• If N < K(K + 1)/2 (respectively N < K2), the Welch bound on

real (respectively, complex) signal sets is the best among the three

bounds. However, the Levenstein bound of (4) on real signal sets

is tighter than Welch’s bound if N > K(K + 1)/2, and that the

Levenstein bound of (5) on complex signal sets is tighter than

Welch’s bound if N > K2. The bound of (6) could be negative,

e.g., when N = 71 and K = 35, and thus makes no sense in

certain cases. But in some cases the lower bound of (6) could be

tighter, compared with the Welch and Levenstein bounds.

J. Yin, Sept. 14, 200668


3.2 Signal Sets from Cyclic DSs

• Define ψN(n) = e2nπ√−1/N . Then ψN is a group character of

(ZN , +). For any K-subset D = {d1, d2, · · · , dK} of ZN , define a

signal set given by

CD = {ci : i ∈ ZN}. (7)

where for each i with 0 ≤ i ≤ N − 1

ci :=1√K

(ψN(id1), ψN(id2), · · · , ψN(idk)) . (8)

J. Yin, Sept. 14, 200669


• Theorem 3.1 (Xia et al. (IEEE IT 2005)) CD is an (N, K)

MWBE signal set if and only if D is a cyclic (N, K, λ)-DS in ZN .

• Theorem 3.1 works only for cyclic difference sets. It is a

generalization of a specific family of equiangular tight frames

(another name for siginal sets) obtained by Konig (Konig 1979).

• Theorem 3.1 produces a number of MWBE signal sets based on

known cyclic difference sets.

J. Yin, Sept. 14, 200670


3.3 Signal Sets from DSs in GF(q)

• Let x0, x1, · · · , xq−1 denote all the elements of the finite field

GF(q) with q = pm. Let εp = e2π√−1/p and Tr(x) be the absolute

trace function from GF(q) to GF(p). For any K-subset

D = {d1, d2, · · · , dK} of GF(q), define a signal set given by

CD = {ci : i = 0, 1, · · · , q − 1}. (9)

where for each i with 0 ≤ i ≤ q − 1

ci :=1√K

(εTr(xid1p , εTr(xid2

p , · · · , εTr(xidKp ,

). (10)

J. Yin, Sept. 14, 200671


• Theorem 3.2 (Ding (IEE IT 2006)) CD is an (N, K) MWBE

signal set if D is a (N,K, λ)-DS in (GF(q), +) where K > 1 .

• Theorem 3.2 works both cyclic and noncyclic difference sets in

GF(q).

J. Yin, Sept. 14, 200672


3.4 Signal sets from PNFs

• Assume that m is a positive integer and p is an odd prime and

q = pm so that there are planar functions (GF(q), +) to

(GF(q), +). As before, use x0, x1, · · · , xq−1 to denote all the

elements of the finite field GF(q). For any positive integer n,

define εn = e2π√−1/n. Let Tr denote the absolute trace function on

GF(q). Define

ψ(x) := εTr(x)p . (11)

Then ψ is an additive group character of GF(q).

J. Yin, Sept. 14, 200673


• Let Π be a perfect nonlinear function from (GF(q), +) to

(GF(q), +). For each pair (a, b) ∈ GF(q)2, define the unit-norm

vector

c(a,b) =1√q

(ψ(aΠ(x0) + bx0), ψ(aΠ(x1) + bx1), ..., ψ(aΠ(xq−1) + bxq−1)) .

Then form a signal set

CΠ = {c(a,b) : (a, b) ∈ GF(q)2}⋃

Eq. (12)

J. Yin, Sept. 14, 200674


• Here, Eq is the standard orthogonal basis of the q-dimensional

Hilbert space, consisting of the following vectors:

e(q)1 = (1, 0, 0, 0, · · · , 0, 0),

e(q)2 = (0, 1, 0, 0, · · · , 0, 0),

......

...

e(q)q = (0, 0, 0, 0, · · · , 0, 1),

J. Yin, Sept. 14, 200675


• Theorem 3.3 (Ding and Yin (IEEE Commun. 2006) CΠ is a

(q2 + q, q) signal set with Imax(CΠ) = 1√q.

J. Yin, Sept. 14, 200676


4.1 Optimal OOCs

• A (v, k, λ1, λ2) optical orthogonal code (briefly

(v, k, λ1, λ2)-OOC), C, is a family of (0, 1)-sequences (codewords)

of length v and weight k satisfying the following two properties:

¨∑

0≤t≤v−1xtxt+i ≤ λ1 for any x = (x0, x1, . . . , xv−1) ∈ C and any

integer i 6≡ 0 mod v (the auto-correlation property) ;

¨∑

0≤t≤v−1xtyt+i ≤ λ2 for any x = (x0, x1, . . . , xv−1) ∈ C,

y = (y0, y1, . . . , yv−1) ∈ C with x 6= y, and any integer i (the

cross-correlation property) .

J. Yin, Sept. 14, 200677


• All subscripts here are reduced modulo v so that only periodic

correlations are considered.

• OOCs are applied in multimedia transmission in fiber-optic LANs

and in multirate fiber-optic CDMA systems which require binary

sequences with good correlation properties.

• Research on (v, k, λ1, λ2)-OOCs in combinatorial community has

mainly concentrated on the case where λ1 = λ2 = 1. In this case,

the CCC is called a (v, k, 1)-OOC, and it is optimal if it contains

b v−1k(k−1)

c codewords.

J. Yin, Sept. 14, 200678


4.2 The Comb. Characterization

• Let F be a family of t k-subsets of Zv. If there exists a (minimum)

constant value λ such that dF(g) ≤ λ for any nonzero residue in

Zv, then F is well known as a cyclic difference packing, or a

(v, k, λ)-CDP in short. When b v−1k(k−1)

c then it is optimal.

• Theorem 4.1 (See, for example, Yin (DM 185 (1998), PP. 201))

An optimal (v, k, 1)-OOC is equivalent to an optimal (v, k, 1)-CDP.

J. Yin, Sept. 14, 200679


• The construction of a (v, k, 1)-CDP with b v−1k(k−1)

c base blocks is a

rather difficult task. The existence of a (v, 4, 1)-CDP with bv−112c

base blocks is far from complete to this day. For the recent results,

the reader may refer to the new version of The CRC Handbook

of Combinatorial Designs

(http://www.cems.uvm.edu/ jcd/hcdproofread/). Its publication

is due this year.

J. Yin, Sept. 14, 200680


5. Splitting A-Codes and EDFs

• Consider a collection F = {D1, D2, · · · , Du} of subsets (base

blocks) of an additive abelian group G of order v. If the base

blocks in F are mutually disjoint, then we have another

difference function, called external difference function, that is

defined as

edF(g) :=∑

1≤i6=j≤u

|(Di + g) ∩Dj|,

for g ∈ G \ {0}.

J. Yin, Sept. 14, 200681


• If edF(g) takes on a constant value λ when g ranges over all the

nonzero elements of G. Then F is referred to as an external

difference family (EDF), or a (v, {k1, k2, · · · , ku}, λ)-EDF in

short. In the case |Di| = k for any i (1 ≤ i ≤ u), it is termed a

(v, k, λ; u)-EDF.

• The notion of an (n, k, λ; u)-EDF of index λ = 1

was first defined by Ogata et al. (DM 2004).

J. Yin, Sept. 14, 200682


• Using such an EDF, Ogata et al. constructed an A-code where the

plaintext (source state) space S = {1, 2, · · · , u}, message space Mand key space E are both taken to be G. For i ∈ S and e ∈ E ,

e(i) = Di. The term “splitting” means that a message is not

uniquely determined by the plaintext and the key, that is very

important in the context of authentication with arbitration. Here

|Di| = k > 1. So the resulting A-code is k-splitting. They also

gave a way to obtain (k, n) secret scheme against cheaters from a

EDFs, and proved that the scheme from such an EDF is optimal

with respect a certain bound.

J. Yin, Sept. 14, 200683


6.1 Comma Free Codes

• The code synchronization problem

In a comma free code, the data stream consists of consecutive

messages each being a sequence of n consecutive symbols. The

synchronization problem that arises at the receiving end is the task

to partition correctly the data stream into messages of length n.

J. Yin, Sept. 14, 200684


Correct decoding:

· · · ︷︸︸︷a1a2 · · · an

︷︸︸︷b1b2 · · · bn · · ·

Incorrect decoding:

︷︸︸︷· · · a1 · · · ai

︷︸︸︷ai+1 · · · anb1 · · · bi

︷︸︸︷· · · bn · · ·

It is the concatenation of the end of one message with the beginningof another message, denoted by Ei(a,b).

J. Yin, Sept. 14, 200685


• One way to resolve the synchronization problem is by requiring

that no codeword c coincides with the concatenation Ei(a,b),

where a and b are two codewords (not necessarily distinct).

• Let (Zq, C) be a (n,M, d; q) code. The comma-free index ρ(C)

of the code C is defined as

ρ(C) = min d(c, Ei(a,b))

where the minimum is taken over all the codewords a,b, c ∈ C

and all i = 1, 2, · · · , n− 1, and d(c, Ei(a,b)) stands for the

Hamming distance.

J. Yin, Sept. 14, 200686


• The comma-free index ρ(C) allows one to distinguish a codeword

from a concatenation of two codewords even in case that up to

b(ρ(C)− 1)/2c errors have occurred (and hence allows

synchronization)(see Golomb et al., 1958).

J. Yin, Sept. 14, 200687


6.2 Comma free codes from DSSs

• Consider a collection F = {D1, D2, · · · , Du} of u disjoint subsets

(base blocks) of Zv. If there exists a (maximum) constant value λ

such that edF(g) ≥ λ for any nonzero residue (mod v). Then F is

called difference system of sets (DSSs) or a

(v, {k1, k2, · · · , ku}, λ)-DSS.

• When |Di| = k for any i (1 ≤ i ≤ u), the DSS is called regular.

As with EDF, we simply say that F is a (v, k, λ; u)-DSS in this

case.

J. Yin, Sept. 14, 200688


• A perfect (v, {k1, k2, · · · , ku}, λ)-DSS is a cyclic EDF.

• Levenshtein (1971, 2004) gave a method to construct codes with

prescribed comma-free index by way of a DSS.

• The detailed can be found in a recent survey paper by Tonchev

(Finite Fields and Their Applications 11 (2005), 601–621).

J. Yin, Sept. 14, 200689


• The application of a DSS of index λ {Dj : j = 0, 1, · · · , q − 1} in

Zn to code synchronization requires the redundancy

rq(n, λ) =

q−1∑j=0

|Dj|

is as small as possible. A DSS is optimal if its redundancy is

equal to rq(n, λ).

J. Yin, Sept. 14, 200690


• Levenshtein (1971) proved that

rq(n, λ) ≥√

λq(n− 1)q − 1

(13)

with the equality if and only if the DSS is perfect and regular.

J. Yin, Sept. 14, 200691


Appendix

The Link between PDFs and EDFs

• PDFs, EDFs and DSSs we have touched upon are different

combinatorial objects, but are closely related. A perfect DSS is

a cyclic EDF. A convenient way to mention the link between

PDFs and EDFs is to use a group ring. Let G be an additive

abelian group and Z the ring of integers. Define the ring of the

formal polynomials over Z of an indeterminate x

Z[x] =

{∑g∈G

agxg : ag ∈ G

}.

J. Yin, Sept. 14, 200692


• The addition of Z[x] is given by∑g∈G

agxg +

∑g∈G

bgxg =

∑g∈G

(ag + bg)xg

• The multiplication of Z[x] is given by(∑

g∈G

agxg

)(∑g∈G

bgxg

)=

∑

h∈G

(∑g∈G

agbh−g

)xh.

J. Yin, Sept. 14, 200693


• The additive unity and the multiplicative unity of Z[x] are defined

respectively as

0 :=∑g∈G

0xg and 1 := x0.

• The Hall polynomial∑

g∈S xg (S ⊆ G) is often simply denoted by

S(x).

J. Yin, Sept. 14, 200694


• Now let F = {D1, D2, · · · , Du} is a partition of an abelian group

G of order v. With the above observations, we can see

¨ F is a (v, {k1, k2, · · · , ku}, λ)-PDF in G provided that

u∑i=1

Di(x)Di(x−1) = v − λ + λG(x).

¨ F is a (v, {k1, k2, · · · , ku}, λ)-EDF in G provided that∑

1≤i 6=j≤u

Di(x)Dj(x−1) = −λ + λG(x).

J. Yin, Sept. 14, 200695


• In summary, F is a (v, {k1, k2, · · · , ku}, λ)-PDF in G if and

only if F is a (v, {k1, k2, · · · , ku}, v − λ)-EDF in G

• A similar relationship between disjoing DFs and external DFs was

mentioned in (Chang and Ding, 2006).

• A DSS is essentially a cyclic external difference covering. We can

establish the link between DSSs and CDPs in a similar vein.

J. Yin, Sept. 14, 200696


THANKS!

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