Denis Krotov SobolevInstituteofMathematics...

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Perfect codes in Doob graphs Denis Krotov Sobolev Institute of Mathematics Novosibirsk, Russia Graphs and Groups, Cycles and Coverings (G2C2) Novosibirsk, 2014

Transcript of Denis Krotov SobolevInstituteofMathematics...

Perfect codes in Doob graphs

Denis KrotovSobolev Institute of Mathematics

Novosibirsk, Russia

Graphs and Groups, Cycles and Coverings (G2C2)Novosibirsk, 2014

Outline

Definitions (Doob graph, perfect codes)

Doob graphs: the structure of a module over the ring GR(42)of Z4.

Linear and additive codes; restrictions on the parameters

Constructions

Outline

Definitions (Doob graph, perfect codes)

Doob graphs: the structure of a module over the ring GR(42)of Z4.

Linear and additive codes; restrictions on the parameters

Constructions

Outline

Definitions (Doob graph, perfect codes)

Doob graphs: the structure of a module over the ring GR(42)of Z4.

Linear and additive codes; restrictions on the parameters

Constructions

Outline

Definitions (Doob graph, perfect codes)

Doob graphs: the structure of a module over the ring GR(42)of Z4.

Linear and additive codes; restrictions on the parameters

Constructions

Distance-regular graphs

A connected regular graph is called distance regular if everybipartite subgraph generated (as parts) by two cocenteredspheres of different radius is biregular.

In other words, a distance-regular graph is a regular graph G

for which there exist integers bi , ci , i = 0, ..., d such that forany two vertices x , y in G and distance i = d(x , y), there areexactly ci neighbors of y in Gi−1(x) and bi neighbors of y inGi+1(x), where Gi (x) is the set of vertices y of G withd(x , y) = i

[Brouwer et al. Distance Regular Graphs. p. 434].

Distance-regular graphs

A connected regular graph is called distance regular if everybipartite subgraph generated (as parts) by two cocenteredspheres of different radius is biregular.

In other words, a distance-regular graph is a regular graph G

for which there exist integers bi , ci , i = 0, ..., d such that forany two vertices x , y in G and distance i = d(x , y), there areexactly ci neighbors of y in Gi−1(x) and bi neighbors of y inGi+1(x), where Gi (x) is the set of vertices y of G withd(x , y) = i

[Brouwer et al. Distance Regular Graphs. p. 434].

Perfect codes

A set of vertices of a graph or any other discrete metric spaceis called an e-perfect code (perfect e-error-correction code),or simply a perfect code, if the vertex set is partitioned intothe radius-e balls centered in the code vertices.

The codes of cardinality 1 and the 0-perfect codes are calledtrivial perfect codes.

Perfect codes

A set of vertices of a graph or any other discrete metric spaceis called an e-perfect code (perfect e-error-correction code),or simply a perfect code, if the vertex set is partitioned intothe radius-e balls centered in the code vertices.

The codes of cardinality 1 and the 0-perfect codes are calledtrivial perfect codes.

Perfect codes in Distance-gerular graphs

The perfect codes in distance regular graphs are objects thatare highly interesting from the point of view of both codingtheory and algebraic combinatorics.

On one hand, these codes are error correcting codes thatattain the sphere-packing bound (“perfect” means “extremelygood”).

On the other hand, they possess algebraic properties that areconnected with the algebraic properties of the distance regulargraph; a perfect code is a some kind of divisor [Cvetkovic etal. Spectra of Graphs: Theory and Application. 1980.Chapter 4] of the graph.

Perfect codes in Distance-gerular graphs

The perfect codes in distance regular graphs are objects thatare highly interesting from the point of view of both codingtheory and algebraic combinatorics.

On one hand, these codes are error correcting codes thatattain the sphere-packing bound (“perfect” means “extremelygood”).

On the other hand, they possess algebraic properties that areconnected with the algebraic properties of the distance regulargraph; a perfect code is a some kind of divisor [Cvetkovic etal. Spectra of Graphs: Theory and Application. 1980.Chapter 4] of the graph.

Perfect codes in Distance-gerular graphs

The perfect codes in distance regular graphs are objects thatare highly interesting from the point of view of both codingtheory and algebraic combinatorics.

On one hand, these codes are error correcting codes thatattain the sphere-packing bound (“perfect” means “extremelygood”).

On the other hand, they possess algebraic properties that areconnected with the algebraic properties of the distance regulargraph; a perfect code is a some kind of divisor [Cvetkovic etal. Spectra of Graphs: Theory and Application. 1980.Chapter 4] of the graph.

It may safely be said that the most important class of distanceregular graphs, for coding theory, is the Hamming graphs.

The Hamming graph H(n, q) is the Cartesian product of ncopies of the complete graph Kq of order q.

It is usual to consider the n-tuples (= words of length n) overthe Galois field GF(q) as the vertices of H(n, q).

The n-tuples form a vector space over GF(q).

A subspace is called a linear code. A linear code can bedefined by a basis (generator matrix) or by a basis of the dualsubspace (check matrix).

It may safely be said that the most important class of distanceregular graphs, for coding theory, is the Hamming graphs.

The Hamming graph H(n, q) is the Cartesian product of ncopies of the complete graph Kq of order q.

It is usual to consider the n-tuples (= words of length n) overthe Galois field GF(q) as the vertices of H(n, q).

The n-tuples form a vector space over GF(q).

A subspace is called a linear code. A linear code can bedefined by a basis (generator matrix) or by a basis of the dualsubspace (check matrix).

It may safely be said that the most important class of distanceregular graphs, for coding theory, is the Hamming graphs.

The Hamming graph H(n, q) is the Cartesian product of ncopies of the complete graph Kq of order q.

It is usual to consider the n-tuples (= words of length n) overthe Galois field GF(q) as the vertices of H(n, q).

The n-tuples form a vector space over GF(q).

A subspace is called a linear code. A linear code can bedefined by a basis (generator matrix) or by a basis of the dualsubspace (check matrix).

It may safely be said that the most important class of distanceregular graphs, for coding theory, is the Hamming graphs.

The Hamming graph H(n, q) is the Cartesian product of ncopies of the complete graph Kq of order q.

It is usual to consider the n-tuples (= words of length n) overthe Galois field GF(q) as the vertices of H(n, q).

The n-tuples form a vector space over GF(q).

A subspace is called a linear code. A linear code can bedefined by a basis (generator matrix) or by a basis of the dualsubspace (check matrix).

It may safely be said that the most important class of distanceregular graphs, for coding theory, is the Hamming graphs.

The Hamming graph H(n, q) is the Cartesian product of ncopies of the complete graph Kq of order q.

It is usual to consider the n-tuples (= words of length n) overthe Galois field GF(q) as the vertices of H(n, q).

The n-tuples form a vector space over GF(q).

A subspace is called a linear code. A linear code can bedefined by a basis (generator matrix) or by a basis of the dualsubspace (check matrix).

Perfect codes in Hamming graphs

For the Hamming graphs H(n, q), the study of possibleparameters of perfect codes is completed only if q is a primepower [Zinoviev, Leontiev, 1973 and Tietavainen, 1973].

1-perfect codes in H( qm−1

q−1, q) (the Hamming codes and the

codes with parameters of Hamming codes)

check matrix:

0 0 0 0 1 1 1 1 1 1 1 1 10 1 1 1 0 0 0 1 1 1 2 2 21 0 1 2 0 1 2 0 1 2 0 1 2

;

3-perfect binary Golay code in H(23, 2);2-perfect ternary Golay code in H(11, 3);binary repetition code in H(2e+1, 2): {000...0, 111...1}

If q is not a prime power (6, 10, 12, 14, 15, . . . ), the problemof existence 1-perfect and (for some q) 2-perfect codes isopen.

Perfect codes in Hamming graphs

For the Hamming graphs H(n, q), the study of possibleparameters of perfect codes is completed only if q is a primepower [Zinoviev, Leontiev, 1973 and Tietavainen, 1973].

1-perfect codes in H( qm−1

q−1, q) (the Hamming codes and the

codes with parameters of Hamming codes)

check matrix:

0 0 0 0 1 1 1 1 1 1 1 1 10 1 1 1 0 0 0 1 1 1 2 2 21 0 1 2 0 1 2 0 1 2 0 1 2

;

3-perfect binary Golay code in H(23, 2);2-perfect ternary Golay code in H(11, 3);binary repetition code in H(2e+1, 2): {000...0, 111...1}

If q is not a prime power (6, 10, 12, 14, 15, . . . ), the problemof existence 1-perfect and (for some q) 2-perfect codes isopen.

Perfect codes in Hamming graphs

For the Hamming graphs H(n, q), the study of possibleparameters of perfect codes is completed only if q is a primepower [Zinoviev, Leontiev, 1973 and Tietavainen, 1973].

1-perfect codes in H( qm−1

q−1, q) (the Hamming codes and the

codes with parameters of Hamming codes)

check matrix:

0 0 0 0 1 1 1 1 1 1 1 1 10 1 1 1 0 0 0 1 1 1 2 2 21 0 1 2 0 1 2 0 1 2 0 1 2

;

3-perfect binary Golay code in H(23, 2);2-perfect ternary Golay code in H(11, 3);binary repetition code in H(2e+1, 2): {000...0, 111...1}

If q is not a prime power (6, 10, 12, 14, 15, . . . ), the problemof existence 1-perfect and (for some q) 2-perfect codes isopen.

Perfect codes in Hamming graphs

For the Hamming graphs H(n, q), the study of possibleparameters of perfect codes is completed only if q is a primepower [Zinoviev, Leontiev, 1973 and Tietavainen, 1973].

1-perfect codes in H( qm−1

q−1, q) (the Hamming codes and the

codes with parameters of Hamming codes)

check matrix:

0 0 0 0 1 1 1 1 1 1 1 1 10 1 1 1 0 0 0 1 1 1 2 2 21 0 1 2 0 1 2 0 1 2 0 1 2

;

3-perfect binary Golay code in H(23, 2);2-perfect ternary Golay code in H(11, 3);binary repetition code in H(2e+1, 2): {000...0, 111...1}

If q is not a prime power (6, 10, 12, 14, 15, . . . ), the problemof existence 1-perfect and (for some q) 2-perfect codes isopen.

Perfect codes in Hamming graphs

For the Hamming graphs H(n, q), the study of possibleparameters of perfect codes is completed only if q is a primepower [Zinoviev, Leontiev, 1973 and Tietavainen, 1973].

1-perfect codes in H( qm−1

q−1, q) (the Hamming codes and the

codes with parameters of Hamming codes)

check matrix:

0 0 0 0 1 1 1 1 1 1 1 1 10 1 1 1 0 0 0 1 1 1 2 2 21 0 1 2 0 1 2 0 1 2 0 1 2

;

3-perfect binary Golay code in H(23, 2);2-perfect ternary Golay code in H(11, 3);binary repetition code in H(2e+1, 2): {000...0, 111...1}

If q is not a prime power (6, 10, 12, 14, 15, . . . ), the problemof existence 1-perfect and (for some q) 2-perfect codes isopen.

Perfect codes in Hamming graphs

For the Hamming graphs H(n, q), the study of possibleparameters of perfect codes is completed only if q is a primepower [Zinoviev, Leontiev, 1973 and Tietavainen, 1973].

1-perfect codes in H( qm−1

q−1, q) (the Hamming codes and the

codes with parameters of Hamming codes)

check matrix:

0 0 0 0 1 1 1 1 1 1 1 1 10 1 1 1 0 0 0 1 1 1 2 2 21 0 1 2 0 1 2 0 1 2 0 1 2

;

3-perfect binary Golay code in H(23, 2);2-perfect ternary Golay code in H(11, 3);binary repetition code in H(2e+1, 2): {000...0, 111...1}

If q is not a prime power (6, 10, 12, 14, 15, . . . ), the problemof existence 1-perfect and (for some q) 2-perfect codes isopen.

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k V } k ¢ k Ó Ô c ù Õ � o � Ö ` Ó © � � ~ É

^ B ¤ × {Ø � B Ù W f Ú ¥ �C

� � ¼�

V | h�

� a ¤ ÷Û Ü á x Ý � � 4 ]

Þ � B | c � Ò K n ` ² a ì ß à á â ã ä H å æ ç è é ê ë ì í î ï ð ñ ò ó ô å ð õ ö ÷ ø î ù ú ûü îý õ þ ÿ � � H � � � �� Y � · a w @ µ U V � � i _   | _ i K ¼ � � � � ï � ï î � ï � H � ù � � � � � � �

¼ ? Å� K � ã g À � { ´ X �

� ��

i � c\

Ì � � � i Ì T !ð

Z � t ·s

@ ² « k " #É

$ �% & ' ( ) ð ï * + , ï - . / 0 1ý ð H 2 3 + ï 4 5 ö 6 78 9: �; < = > ? ? ó = @ ABC � D g i Ô ~ U a � E u i F � a g i G � | H I J K L ï 2 � ï î ð ð M îN O P Q ð R ý S ð � å � T UN É H V ï U ð � 4 W X � Y Z[ \ ] ^ _ ` a b c d e f g _ h i j k l f m n o p q r s t u v w x y z { õ | } ~ � � � H � � � � ñ � � U U H ï� � . � � � � � � � � � � � � � � � � � � � � � � � � � � � w   ¡ ¢ £ ¤ ¥ é ¦ § ¨ . © ª « ¬ ­ ® ¯ � � U ;° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á  ± Ã Ä Å Æ � Ç È É Ê Ë Ì Í ¡ Î Ï Ð Ñ Ò Ó Ï Ô Õ Ö ×Ø Ù Ú · Û « ï Ü « Ý « Þ î � å åß u à á â � ã ä å æ ç è é ê ëì íî ïð ñ ò ó ô

õ ö ÷ ø ùú

û ü ý þ ÿ �

The Doob graphs

The Doob graph D(m, n) is a distance regular graph ofdiameter 2m + n with the same parameters as the Hamminggraph H(2m + n, 4).

The Doob graph D(m, n) is the Cartesian product of m copiesof the Shrikhande graph Sh and n copies of the full graphK = K4 of order 4.

The Doob graphs

The Doob graph D(m, n) is a distance regular graph ofdiameter 2m + n with the same parameters as the Hamminggraph H(2m + n, 4).

The Doob graph D(m, n) is the Cartesian product of m copiesof the Shrikhande graph Sh and n copies of the full graphK = K4 of order 4.

Perfect codes in Doob graphs

As noted in [Koolen, Munemasa, 2000], nontrivial e-perfectcodes in D(m, n) can only exist when e = 1 and2m + n = (4µ − 1)/3 for some integer µ (with exactly thesame proof as for H(2m + n, 4)).

Koolen and Munemasa constructed 1-perfect codes in theDoob graphs of diameter 5 (i.e., D(1, 3) and D(2, 1)).

In the current work:

Restrictions on the parameters of linear 1-perfect codes inD(m, n)

Construction of linear codes in D(m, n) with the structure of(GR(42))m × (GF(22))n.

Construction of linear codes in D(m, n′ + n′′) with thestructure of (Z2

4)m × (Z2

2)n′ × Z

n′′

4 .

Perfect codes in Doob graphs

As noted in [Koolen, Munemasa, 2000], nontrivial e-perfectcodes in D(m, n) can only exist when e = 1 and2m + n = (4µ − 1)/3 for some integer µ (with exactly thesame proof as for H(2m + n, 4)).

Koolen and Munemasa constructed 1-perfect codes in theDoob graphs of diameter 5 (i.e., D(1, 3) and D(2, 1)).

In the current work:

Restrictions on the parameters of linear 1-perfect codes inD(m, n)

Construction of linear codes in D(m, n) with the structure of(GR(42))m × (GF(22))n.

Construction of linear codes in D(m, n′ + n′′) with thestructure of (Z2

4)m × (Z2

2)n′ × Z

n′′

4 .

Perfect codes in Doob graphs

As noted in [Koolen, Munemasa, 2000], nontrivial e-perfectcodes in D(m, n) can only exist when e = 1 and2m + n = (4µ − 1)/3 for some integer µ (with exactly thesame proof as for H(2m + n, 4)).

Koolen and Munemasa constructed 1-perfect codes in theDoob graphs of diameter 5 (i.e., D(1, 3) and D(2, 1)).

In the current work:

Restrictions on the parameters of linear 1-perfect codes inD(m, n)

Construction of linear codes in D(m, n) with the structure of(GR(42))m × (GF(22))n.

Construction of linear codes in D(m, n′ + n′′) with thestructure of (Z2

4)m × (Z2

2)n′ × Z

n′′

4 .

Perfect codes in Doob graphs

As noted in [Koolen, Munemasa, 2000], nontrivial e-perfectcodes in D(m, n) can only exist when e = 1 and2m + n = (4µ − 1)/3 for some integer µ (with exactly thesame proof as for H(2m + n, 4)).

Koolen and Munemasa constructed 1-perfect codes in theDoob graphs of diameter 5 (i.e., D(1, 3) and D(2, 1)).

In the current work:

Restrictions on the parameters of linear 1-perfect codes inD(m, n)

Construction of linear codes in D(m, n) with the structure of(GR(42))m × (GF(22))n.

Construction of linear codes in D(m, n′ + n′′) with thestructure of (Z2

4)m × (Z2

2)n′ × Z

n′′

4 .

Perfect codes in Doob graphs

As noted in [Koolen, Munemasa, 2000], nontrivial e-perfectcodes in D(m, n) can only exist when e = 1 and2m + n = (4µ − 1)/3 for some integer µ (with exactly thesame proof as for H(2m + n, 4)).

Koolen and Munemasa constructed 1-perfect codes in theDoob graphs of diameter 5 (i.e., D(1, 3) and D(2, 1)).

In the current work:

Restrictions on the parameters of linear 1-perfect codes inD(m, n)

Construction of linear codes in D(m, n) with the structure of(GR(42))m × (GF(22))n.

Construction of linear codes in D(m, n′ + n′′) with thestructure of (Z2

4)m × (Z2

2)n′ × Z

n′′

4 .

Perfect codes in Doob graphs

As noted in [Koolen, Munemasa, 2000], nontrivial e-perfectcodes in D(m, n) can only exist when e = 1 and2m + n = (4µ − 1)/3 for some integer µ (with exactly thesame proof as for H(2m + n, 4)).

Koolen and Munemasa constructed 1-perfect codes in theDoob graphs of diameter 5 (i.e., D(1, 3) and D(2, 1)).

In the current work:

Restrictions on the parameters of linear 1-perfect codes inD(m, n)

Construction of linear codes in D(m, n) with the structure of(GR(42))m × (GF(22))n.

Construction of linear codes in D(m, n′ + n′′) with thestructure of (Z2

4)m × (Z2

2)n′ × Z

n′′

4 .

Eisenstein integers

The Eisenstein integers E are the complex numbers of theform

z = a+ bω, ω =−1 + i

√3

2= e2πi/3, a, b ∈ Z.

Re

Im

−2−2 00 22 44−1−1 11 33

2ω+12ω+12ω2ω

2ω2+12ω2+1

−2+ω−2+ω −1+ω−1+ω ωω −ω2−ω2

ω2ω2 −ω−ω

Eisenstein integers modulo 2 or 4

Given p ∈ E\{0}, we denote by Ep the ring E/pE of residueclasses of E modulo p.

We are interested in: E2 ≃ GF(22) and E4 ≃ GR(42)

Each of 16 elements of E4 can be uniquely represented in theform 2a+ b where a, b ∈ {0, 1, ω, ω2}.E = {−ω2, ω,−1, ω2,−ω, 1} is the set of units.

The elements of E4 are partitioned into 0E , E , 2E , and ψEwhere ψ = 2 + ω.

Eisenstein integers modulo 2 or 4

Given p ∈ E\{0}, we denote by Ep the ring E/pE of residueclasses of E modulo p.

We are interested in: E2 ≃ GF(22) and E4 ≃ GR(42)

Each of 16 elements of E4 can be uniquely represented in theform 2a+ b where a, b ∈ {0, 1, ω, ω2}.E = {−ω2, ω,−1, ω2,−ω, 1} is the set of units.

The elements of E4 are partitioned into 0E , E , 2E , and ψEwhere ψ = 2 + ω.

Eisenstein integers modulo 2 or 4

Given p ∈ E\{0}, we denote by Ep the ring E/pE of residueclasses of E modulo p.

We are interested in: E2 ≃ GF(22) and E4 ≃ GR(42)

Each of 16 elements of E4 can be uniquely represented in theform 2a+ b where a, b ∈ {0, 1, ω, ω2}.E = {−ω2, ω,−1, ω2,−ω, 1} is the set of units.

The elements of E4 are partitioned into 0E , E , 2E , and ψEwhere ψ = 2 + ω.

Eisenstein integers modulo 2 or 4

Given p ∈ E\{0}, we denote by Ep the ring E/pE of residueclasses of E modulo p.

We are interested in: E2 ≃ GF(22) and E4 ≃ GR(42)

Each of 16 elements of E4 can be uniquely represented in theform 2a+ b where a, b ∈ {0, 1, ω, ω2}.E = {−ω2, ω,−1, ω2,−ω, 1} is the set of units.

The elements of E4 are partitioned into 0E , E , 2E , and ψEwhere ψ = 2 + ω.

Eisenstein integers modulo 2 or 4

Given p ∈ E\{0}, we denote by Ep the ring E/pE of residueclasses of E modulo p.

We are interested in: E2 ≃ GF(22) and E4 ≃ GR(42)

Each of 16 elements of E4 can be uniquely represented in theform 2a+ b where a, b ∈ {0, 1, ω, ω2}.E = {−ω2, ω,−1, ω2,−ω, 1} is the set of units.

The elements of E4 are partitioned into 0E , E , 2E , and ψEwhere ψ = 2 + ω.

The Shrikhande graph

Re

Im

22 00 22 0033 11 33

2ω2+12ω2+1 2ω+12ω+1 2ω2+12ω2+1 2ω+12ω+12ω2ω 2ω22ω2 2ω2ω

2ω+12ω+1 2ω2+12ω2+1 2ω+12ω+1 2ω2+12ω2+12ω22ω2 2ω2ω 2ω22ω2

2+ω2+ω 2ω+ω22ω+ω2 ωω 3ω23ω2 2+ω2+ω 2ω+ω22ω+ω2 ωω

2+ω22+ω2 2ω2+ω2ω2+ω ω2ω2 3ω3ω 2+ω22+ω2 2ω2+ω2ω2+ω ω2ω2

The Shrikhande graph is the Cayley graph of E4 with thegenerating set E .

The full graph. Doob graphs as modules. Linear codes.

Similarly, the vertices of K = K4 can be treated as theelements 0, 1, ω, ω2 of E2 ≃ GF(22).

Then, the vertices of D(m, n) are the elements of Em4 × E

n2,

which is a module over the ring E4.

A sumbodule of Em4 × E

n2 is then a linear code in D(m, n).

Any weight-1 word (i.e., adjacent to 0) has an element of E inthe first m coordinates or an element of {1, ω, ω2} in the lastn coordinates.

The full graph. Doob graphs as modules. Linear codes.

Similarly, the vertices of K = K4 can be treated as theelements 0, 1, ω, ω2 of E2 ≃ GF(22).

Then, the vertices of D(m, n) are the elements of Em4 × E

n2,

which is a module over the ring E4.

A sumbodule of Em4 × E

n2 is then a linear code in D(m, n).

Any weight-1 word (i.e., adjacent to 0) has an element of E inthe first m coordinates or an element of {1, ω, ω2} in the lastn coordinates.

The full graph. Doob graphs as modules. Linear codes.

Similarly, the vertices of K = K4 can be treated as theelements 0, 1, ω, ω2 of E2 ≃ GF(22).

Then, the vertices of D(m, n) are the elements of Em4 × E

n2,

which is a module over the ring E4.

A sumbodule of Em4 × E

n2 is then a linear code in D(m, n).

Any weight-1 word (i.e., adjacent to 0) has an element of E inthe first m coordinates or an element of {1, ω, ω2} in the lastn coordinates.

The full graph. Doob graphs as modules. Linear codes.

Similarly, the vertices of K = K4 can be treated as theelements 0, 1, ω, ω2 of E2 ≃ GF(22).

Then, the vertices of D(m, n) are the elements of Em4 × E

n2,

which is a module over the ring E4.

A sumbodule of Em4 × E

n2 is then a linear code in D(m, n).

Any weight-1 word (i.e., adjacent to 0) has an element of E inthe first m coordinates or an element of {1, ω, ω2} in the lastn coordinates.

Doob graphs as modules over Z4. Additive codes.

E4 itself is a module over Z4; its elements are represented bythe pairs from Z

24, in the basis (ω, 1).

Similarly, E2 −→ Z22.

Alternatively, the vertices of K4 can be represented byelements of Z4.

Then, the vertices of D(m, n′ + n′′) are the elements of themodule (Z2

4)m × (Z2

2)n′ × Z

n′′

4 over the ring Z4.

A sumbodule C of M = (Z24)

m × (Z22)

n′ × Zn′′

4 is then anadditive code of type (n′, n′′,m; Γ,∆) in D(m, n′ + n′′).Where M/C ≃ Z

Γ2 × Z

∆4 .

Any weight-1 words has values from {01, 10, 11, 03, 30, 33} ina pair from the first 2m coordinates or values from{01, 10, 11} in a pair from the next 2n′ coordinates or anelement of {1, 2, 3} in the last n′′ coordinates.

Doob graphs as modules over Z4. Additive codes.

E4 itself is a module over Z4; its elements are represented bythe pairs from Z

24, in the basis (ω, 1).

Similarly, E2 −→ Z22.

Alternatively, the vertices of K4 can be represented byelements of Z4.

Then, the vertices of D(m, n′ + n′′) are the elements of themodule (Z2

4)m × (Z2

2)n′ × Z

n′′

4 over the ring Z4.

A sumbodule C of M = (Z24)

m × (Z22)

n′ × Zn′′

4 is then anadditive code of type (n′, n′′,m; Γ,∆) in D(m, n′ + n′′).Where M/C ≃ Z

Γ2 × Z

∆4 .

Any weight-1 words has values from {01, 10, 11, 03, 30, 33} ina pair from the first 2m coordinates or values from{01, 10, 11} in a pair from the next 2n′ coordinates or anelement of {1, 2, 3} in the last n′′ coordinates.

Doob graphs as modules over Z4. Additive codes.

E4 itself is a module over Z4; its elements are represented bythe pairs from Z

24, in the basis (ω, 1).

Similarly, E2 −→ Z22.

Alternatively, the vertices of K4 can be represented byelements of Z4.

Then, the vertices of D(m, n′ + n′′) are the elements of themodule (Z2

4)m × (Z2

2)n′ × Z

n′′

4 over the ring Z4.

A sumbodule C of M = (Z24)

m × (Z22)

n′ × Zn′′

4 is then anadditive code of type (n′, n′′,m; Γ,∆) in D(m, n′ + n′′).Where M/C ≃ Z

Γ2 × Z

∆4 .

Any weight-1 words has values from {01, 10, 11, 03, 30, 33} ina pair from the first 2m coordinates or values from{01, 10, 11} in a pair from the next 2n′ coordinates or anelement of {1, 2, 3} in the last n′′ coordinates.

Doob graphs as modules over Z4. Additive codes.

E4 itself is a module over Z4; its elements are represented bythe pairs from Z

24, in the basis (ω, 1).

Similarly, E2 −→ Z22.

Alternatively, the vertices of K4 can be represented byelements of Z4.

Then, the vertices of D(m, n′ + n′′) are the elements of themodule (Z2

4)m × (Z2

2)n′ × Z

n′′

4 over the ring Z4.

A sumbodule C of M = (Z24)

m × (Z22)

n′ × Zn′′

4 is then anadditive code of type (n′, n′′,m; Γ,∆) in D(m, n′ + n′′).Where M/C ≃ Z

Γ2 × Z

∆4 .

Any weight-1 words has values from {01, 10, 11, 03, 30, 33} ina pair from the first 2m coordinates or values from{01, 10, 11} in a pair from the next 2n′ coordinates or anelement of {1, 2, 3} in the last n′′ coordinates.

Doob graphs as modules over Z4. Additive codes.

E4 itself is a module over Z4; its elements are represented bythe pairs from Z

24, in the basis (ω, 1).

Similarly, E2 −→ Z22.

Alternatively, the vertices of K4 can be represented byelements of Z4.

Then, the vertices of D(m, n′ + n′′) are the elements of themodule (Z2

4)m × (Z2

2)n′ × Z

n′′

4 over the ring Z4.

A sumbodule C of M = (Z24)

m × (Z22)

n′ × Zn′′

4 is then anadditive code of type (n′, n′′,m; Γ,∆) in D(m, n′ + n′′).Where M/C ≃ Z

Γ2 × Z

∆4 .

Any weight-1 words has values from {01, 10, 11, 03, 30, 33} ina pair from the first 2m coordinates or values from{01, 10, 11} in a pair from the next 2n′ coordinates or anelement of {1, 2, 3} in the last n′′ coordinates.

Doob graphs as modules over Z4. Additive codes.

E4 itself is a module over Z4; its elements are represented bythe pairs from Z

24, in the basis (ω, 1).

Similarly, E2 −→ Z22.

Alternatively, the vertices of K4 can be represented byelements of Z4.

Then, the vertices of D(m, n′ + n′′) are the elements of themodule (Z2

4)m × (Z2

2)n′ × Z

n′′

4 over the ring Z4.

A sumbodule C of M = (Z24)

m × (Z22)

n′ × Zn′′

4 is then anadditive code of type (n′, n′′,m; Γ,∆) in D(m, n′ + n′′).Where M/C ≃ Z

Γ2 × Z

∆4 .

Any weight-1 words has values from {01, 10, 11, 03, 30, 33} ina pair from the first 2m coordinates or values from{01, 10, 11} in a pair from the next 2n′ coordinates or anelement of {1, 2, 3} in the last n′′ coordinates.

Parameters of additive 1-perfect codes.

Theorem

Assume there is an additive 1-perfect code C of type

(n′, n′′,m; Γ,∆) in D(m, n′ + n′′). Then n′′ 6= 1, Γ is even,

2m + n′ + n′′ = (2Γ+2∆ − 1)/3, (1)

3n′ + n′′ = 2Γ+∆ − 1, (2)

n′′ ≤ 2∆ − 1 (3)

The number of elements in the factor-group is the number ofweight≤ 1 words. (1) follows. Then Γ is even.

The number of order-2 elements in the factor-group is thenumber of weight-1 words of order-2. (2) follows.

There are at least n′′ weight-1 words of order 2 that aremultiples of 2. (3) follows.

n′′ 6= 1 follows from the fact that a projective space cannot bepartitioned into one point and several lines.

Parameters of additive 1-perfect codes.

Theorem

Assume there is an additive 1-perfect code C of type

(n′, n′′,m; Γ,∆) in D(m, n′ + n′′). Then n′′ 6= 1, Γ is even,

2m + n′ + n′′ = (2Γ+2∆ − 1)/3, (1)

3n′ + n′′ = 2Γ+∆ − 1, (2)

n′′ ≤ 2∆ − 1 (3)

The number of elements in the factor-group is the number ofweight≤ 1 words. (1) follows. Then Γ is even.

The number of order-2 elements in the factor-group is thenumber of weight-1 words of order-2. (2) follows.

There are at least n′′ weight-1 words of order 2 that aremultiples of 2. (3) follows.

n′′ 6= 1 follows from the fact that a projective space cannot bepartitioned into one point and several lines.

Parameters of additive 1-perfect codes.

Theorem

Assume there is an additive 1-perfect code C of type

(n′, n′′,m; Γ,∆) in D(m, n′ + n′′). Then n′′ 6= 1, Γ is even,

2m + n′ + n′′ = (2Γ+2∆ − 1)/3, (1)

3n′ + n′′ = 2Γ+∆ − 1, (2)

n′′ ≤ 2∆ − 1 (3)

The number of elements in the factor-group is the number ofweight≤ 1 words. (1) follows. Then Γ is even.

The number of order-2 elements in the factor-group is thenumber of weight-1 words of order-2. (2) follows.

There are at least n′′ weight-1 words of order 2 that aremultiples of 2. (3) follows.

n′′ 6= 1 follows from the fact that a projective space cannot bepartitioned into one point and several lines.

Parameters of additive 1-perfect codes.

Theorem

Assume there is an additive 1-perfect code C of type

(n′, n′′,m; Γ,∆) in D(m, n′ + n′′). Then n′′ 6= 1, Γ is even,

2m + n′ + n′′ = (2Γ+2∆ − 1)/3, (1)

3n′ + n′′ = 2Γ+∆ − 1, (2)

n′′ ≤ 2∆ − 1 (3)

The number of elements in the factor-group is the number ofweight≤ 1 words. (1) follows. Then Γ is even.

The number of order-2 elements in the factor-group is thenumber of weight-1 words of order-2. (2) follows.

There are at least n′′ weight-1 words of order 2 that aremultiples of 2. (3) follows.

n′′ 6= 1 follows from the fact that a projective space cannot bepartitioned into one point and several lines.

Construction of linear codes using check matrices

A check matrix of a linear code in D(m, n) consists of two parts:

A = Aγ,δ = A∗|A′

with elements from E4 and E2, respectively. We define themultiplication AzT for z = (x |y) as

A∗xT + 2A′yT

(here, the result of the multiplication by 2 is considered as acolumn-vector over E4).

Check matrices of linear 1-perfect codes

A0,2 =

(

0 0 2 2 2ω 2ω 2ω2 2ω2 1 1 1 1 1 11 ψ 1 ψ 1 ψ 1 ψ 0 2 2ω 2ω2 1 −ω1 1 ψ . . . ψ 0 1 1 1 1

2ω2+1 2+ω2 0 . . . 2+ω2 1 0 1 ω ω2

)

A1,1 =

(

1 1 1 1 ψ ψ ψ ψ 0 1 1 1 1

0 2 2ω 2ω2 0 2 2ω 2ω2 1 0 1 ω ω2

)

A0,1 =(

1 ψ 1)

Characterization of linear codes

Theorem (Theorem 2)

Linear 1-perfect codes in D(m, n) exist if and only if for some

integer γ ≥ 0 and δ > 0,

n = (4γ+δ − 1)/3 and m = (4γ+2δ − 4γ+δ)/6.

From linear codes to additive codes

From linear 1-perfect codes we trivially get 1-perfect additivecodes of type (n, 0,m; 2γ, 2δ).

Then, we can obtain 1-perfect additive codes of type(n − n′′/3, n′′,m − n′′/3; 2γ, 2δ) with nonzero n′′. The idea isthe following:

Assume the same column h occurs in both E4 and E2 parts ofthe check matrix. Then, it “cover” the syndromesh, ωh, ω2h, 3h, 3ωh, 3ω2h and 2h, 2ωh, 2ω2h. Regroupinggives h, 2h, 3h, and ωh, 2ωh, 3ωh, and ω2h, 2ω2h, 3ω2h, whichcan be “covered” by three Z4 coordinates in the n′′-part.

(

. . 1 . . . . 1 . .

. . ω2 . . . . ω2 . .

)

−→

. . 1 0 . . . . 1 0 . .

. . 0 1 . . . . 0 1 . .

. . 0 3 . . . . 0 1 . .

. . 1 3 . . . . 1 1 . .

−→

. . . . . . 0 1 3 . .

. . . . . . 1 0 3 . .

. . . . . . 3 0 1 . .

. . . . . . 3 1 0 . .

Additive codes

Theorem (Theorem 3)

Additive 1-perfect codes exist for all parameters that satisfy the

conclusion of Theorem 1 with even ∆.

PROBLEM! Construct additive 1-perfect codes with odd δ

Additive codes, δ = 3

Check matrix for an additive 1-perfect code of type (0, 7, 7; 0, 3):

1 2 2 2 0 3 3 2 0 3 1 3 1 1 1 0 0 1 2 3 10 3 3 0 2 3 1 1 3 3 3 0 0 2 0 1 0 3 3 3 22 2 0 3 3 2 0 3 1 3 1 1 1 2 0 0 1 2 3 1 1

The columns of the matrix are considered as vectors over Z4 thatrepresent elements of the Galois ring GR(43). Let ξ be a primitiveseventh root of 1 in GR(43). The first 14 columns of the matrixare divided into the pairs ξi + 2ξi+2, ξi+1 + 2ξi+5, i = 0, 1, . . . , 6;the last 7 columns are ξ0, ξ1, . . . , ξ6.

Non-linear codes

Theorem (Theorem 4)

Assume that positive integers m, n, µ satisfy

2m + n = (4µ − 1)/3,

m ≤{

(4µ − 2.5 · 2µ + 1)/9 if µ is odd,

(4µ − 2 · 2µ + 1)/9 if µ is even.(4)

Then there is a 1-perfect code in the Doob graph D(m, n).

Problems

For every value (m, n) satisfying 2m+ n = (4µ − 1)/3 and notcovered by the constructions, construct a 1-perfect code inD(m, n) or prove its nonexistence. In particular, do there exist1-perfect codes in D(6, 9), D(9, 3), D(10, 1)?

For every value (n′, n′′,m) satisfying (1)–(3) with odd ∆ ≥ 3(except the case (0, 7, 7)), construct an additive 1-perfectcode of type (n′, n′′,m; Γ,∆) in the D(m, n′ + n′′) or prove itsnonexistence. In particular, does there exist an additive1-perfect code of type (1, 4, 8; 0, 3) in D(8, 5)?

Are the constructed non-additive codes (Theorem 4)propelinear?