Characterizing Finite Frobenius Rings Via Coding Theory
Transcript of Characterizing Finite Frobenius Rings Via Coding Theory
Characterizing Finite Frobenius RingsVia Coding Theory
Jay A. Wood
Department of MathematicsWestern Michigan University
http://homepages.wmich.edu/∼jwood/
Algebra and Communications SeminarUniversity College Dublin
November 7, 2011
Florence Jessie MacWilliams
I 1917–1990
I Bell Labs
I 1962 Harvard dissertation under Andrew Gleason:“Combinatorial Problems of Elementary AbelianGroups”
I Three sections:I Extension theorem on isometriesI The MacWilliams identitiesI Coverings
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Linear Codes Defined over Finite Rings
I Let R be a finite ring with 1. A linear code of lengthn defined over R is a left R-submodule C ⊂ Rn.
I There were some results on codes over rings in the1970s, but the real breakthrough came in 1994.Hammons, Kumar, Calderbank, Sloane, and Soleshowed that important duality properties of certainnon-linear binary codes could be explained by linearcodes defined over Z/4Z.
I Are the fundamental results of MacWilliams validover finite rings?
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Code Equivalence
I When should two linear codes be considered thesame?
I Monomial equivalence (external)
I Linear isometries (internal)
I These notions are the same over finite fields: theMacWilliams extension theorem.
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Monomial equivalence
I Work over a finite ring R .
I A permutation σ of {1, . . . , n} and invertibleelements (units) u1, . . . , un in R determine amonomial transformation T : Rn → Rn by
T (x1, . . . , xn) = (xσ(1)u1, . . . , xσ(n)un).
I Two linear codes C1,C2 ⊂ Rn are monomiallyequivalent if there exists a monomial transformationT such that C2 = T (C1).
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Linear Isometries
I The Hamming weight wt(x) of a vectorx = (x1, . . . , xn) ∈ Rn is the number of nonzeroentries in x .
I A linear isomorphism f : C1 → C2 between linearcodes C1,C2 ⊂ Rn is an isometry if it preservesHamming weight: wt(f (x)) = wt(x), for all x ∈ C1.
I If T is a monomial transformation with C2 = T (C1),then the restriction of T to C1 is an isometry.
I Is the converse true? Does every linear isometrycome from a monomial transformation?
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MacWilliams Extension Theorem overFinite Fields
Assume C1,C2 are linear codes in Fnq. If a linear
isomorphism f : C1 → C2 preserves Hamming weight,then f extends to a monomial transformation of Fn
q.
I MacWilliams (1961); Bogart, Goldberg, Gordon(1978)
I Ward, Wood (1996)
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Generalizing the Work of MacWilliams
I When A = R , is the MacWilliams extension theoremstill valid?
I Yes, if R is a finite Frobenius ring.
I Why Frobenius?
I There is a character-theoretic proof over finite fieldsthat uses the crucial property F ∼= F.
I Frobenius rings satisfy R ∼= R , and the same proofwill work.
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Characters of Finite Abelian Groups
I Let (G ,+) be a finite abelian group.
I A character π of G is a group homomorphismπ : (G ,+)→ (C×,×), where (C×,×) is themultiplicative group of nonzero complex numbers.
I Example: let G = Z/nZ be the integers modulo n.For any a ∈ Z/nZ, πa(x) = exp(2πiax/n), x ∈ G , isa character of G .
I Example: let G = Fq. For any a ∈ Fq,πa(x) = exp(2πi Tr(ax)/p), x ∈ Fq, is a characterof Fq. (Tr : Fq → Fp is the absolute trace to theprime subfield.)
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Character Groups
I The set G of all characters of G is itself a finiteabelian group called the character group.
I |G | = |G |.I As elements of the vector space of all functions from
G to C, the characters are linearly independent.
I If M is a finite left R-module, then M is a rightR-module.
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Two Useful Formulas
∑x∈G
π(x) =
{|G |, π = 1,
0, π 6= 1.
∑π∈G
π(x) =
{|G |, x = 0,
0, x 6= 0.
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Finite Frobenius Rings
I Finite ring R with 1.
I The (Jacobson) radical Rad(R) of R is theintersection of all maximal left ideals of R ; Rad(R)is a two-sided ideal of R .
I The (left/right) socle Soc(R) of R is the ideal of Rgenerated by all the simple left/right ideals of R .
I R is Frobenius if R/Rad(R) ∼= Soc(R) as one-sidedmodules (both left and right).
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Two Useful Theorems About FiniteFrobenius Rings
I (Honold, 2001) R/Rad(R) ∼= Soc(RR) as leftmodules iff R/Rad(R) ∼= Soc(RR) as right modules.
I R is Frobenius iff R ∼= R as left modules iff R ∼= Ras right modules (Hirano, 1997; indep. 1999).
I Corollary: R is Frobenius iff there exists a characterπ of R such that ker π contains no nonzero left(right) ideal of R . This π is a generating character.
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Examples of Finite Frobenius Rings
I Finite fields Fq: π(x) = exp(2πi Tr(x)/p).
I Z/nZ: π(x) = exp(2πix/n).
I Galois rings (Galois extensions of Z/pmZ).
I Finite chain rings (all ideals form a chain).
I Products of Frobenius rings.
I Matrix rings over a Frobenius ring: Mn(R).
I Finite group rings over a Frobenius ring: R[G ].
I F2[X ,Y ]/(X 2,XY ,Y 2) is not Frobenius (Klemm,1989).
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MacWilliams Extension Theorem overFinite Frobenius Rings
Theorem (1999)Let R be a finite Frobenius ring, and supposeC1,C2 ⊂ Rn are left linear codes. If f : C1 → C2 is anR-linear isomorphism that preserves Hamming weight,then f extends to a monomial transformation of Rn.
I Also, Greferath and Schmidt (2000), using posettechniques.
I Greferath (2002), generalizing Bogart, et al.
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Character-Theoretic Proof (a)
I The proof follows a proof of Ward and Wood in thefinite field case (1996).
I View Ci as the image of Λi : M → Rn, withΛi = (λi ,1, . . . , λi ,n) and Λ2 = f ◦ Λ1.
I Using character sums, express Hamming weight as:
wt(Λi(x)) = n −n∑
j=1
1
|R |∑π∈R
π(λi ,j(x)), x ∈ M .
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Character-Theoretic Proof (b)
I Because f preserves Hamming weight, we get
n∑j=1
∑π∈R
π(λ1,j(x)) =n∑
k=1
∑ψ∈R
ψ(λ2,k(x)), x ∈ M .
I In a Frobenius ring, there is a generating characterρ. Every character of R has the form aρ, a ∈ R .
I (aρ)(r) := ρ(ra), r ∈ R .
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Character-Theoretic Proof (c)
I Re-write weight-preservation equation as
n∑j=1
∑a∈R
(aρ)(λ1,j(x)) =n∑
k=1
∑b∈R
(bρ)(λ2,k(x)), x ∈ M .
I Or as
n∑j=1
∑a∈R
ρ(λ1,j(x)a) =n∑
k=1
∑b∈R
ρ(λ2,k(x)b), x ∈ M .
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Character-Theoretic Proof (d)
I The last equation is an equation of characters on M .
I Characters are linearly independent, so one canmatch up terms (carefully).
I A technical argument involving a preordering givenby divisibility in R shows how to match up termswith units as multipliers.
I This produces a permutation σ and units ui in Rsuch that λ2,k = λ1,σ(k)uk , as desired.
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Re-Write the Extension Problem
I The character-theoretic proof just given generalizedthe Ward-Wood proof over finite fields.
I Now we will generalize an approach due toMacWilliams; Bogart, Goldberg, and Gordon; andGreferath in order to re-formulate the extensionproblem.
I Will use R-linear codes over an alphabet A, an ideaof Nechaev and his collaborators.
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Monomial Transformations
I R finite ring, A finite left R-module (an alphabet).
I A linear code over A is a left R-submodule C ⊂ An.
I A monomial transformation T : An → An has theform
T (x1, . . . , xn) = (xσ(1)φ1, . . . , xσ(n)φn),
for (x1, . . . , xn) ∈ An, where σ is a permutation of{1, . . . , n} and φ1, . . . , φn ∈ Aut(A).
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Re-Formulation of Extension Problem (a)
I View a left R-linear code C ⊂ An as the image of anR-linear homomorphism Λ : M → An, whereΛ = (λ1, . . . , λn) and λi : M → A are R-linear.
I Up to monomial equivalence, what matters is thenumber of λi ’s in a given scale class (under rightaction by automorphisms of A).
I The group Aut(A) of R-automorphisms of A acts onthe right on the group HomR(M ,A) of R-linearhomomorphisms from M to A.
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Re-Formulation of Extension Problem (b)
I Let O] be the set of nonzero orbits of the action ofAut(A) on HomR(M ,A).
I Let η : O] → N be the multiplicity function thatcounts how many of the λi belong to each scaleclass.
I Functions equivalent to η have appeared elsewhereunder various names (value function, multiset, etc.).
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Re-Formulation of Extension Problem (c)
I Summary, so far: the monomial equivalence class ofΛ : M → An is encoded by its multiplicity functionη : O] → N.
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Re-Formulation of Extension Problem (d)
I Now, turn to Hamming weights.
I Note that the Hamming weight depends only on theleft scale class of x ∈ M via units of R :
wt(Λ(ux)) = wt(uΛ(x)) = wt(Λ(x)), x ∈ M , u ∈ U .
I Let O be the set of nonzero orbits of the left actionof the group of units U on M .
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Re-Formulation of Extension Problem (e)
I The Hamming weight wt(Λ(x)) depends only on thescale classes of the λi (φi ∈ Aut(A)):
wt(Λ(x)) =n∑
i=1
wt(λi(x)) =n∑
i=1
wt(λi(x)φi).
I The Hamming weight does not depend on the orderof the λi .
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Re-Formulation of Extension Problem (f)
I Let F (O],N) denote the set of all functions fromO] to N. Similarly for F (O,N).
I The Hamming weight gives a well-defined mapW : F (O],N)→ F (O,N):
W (η)(x) =∑λ∈O]
η(λ) wt(λ(x)).
I Summary: the Extension Theorem for Hammingweight holds iff the map W is injective for everyfinite module M .
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Re-Formulation of Extension Problem (g)
I By formally allowing rational coefficients, we get
W : F (O],Q)→ F (O,Q).
I W is a linear transformation of Q-vector spaces.
I The Extension Theorem for Hamming weight holdsiff the map W is injective for every finite module M .
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A Counter-Example to Extension (a)
I For R-linear codes defined over a module A, theextension theorem might not hold.
I Let R = Mm(Fq), the ring of m ×m matrices overFq. The group of units is U = GL(m,Fq).
I Let A = Mm,k(Fq), the space of all m × k matrices.A is a left R-module. Aut(A) = GL(k ,Fq).
I Assume m < k .
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A Counter-Example to Extension (b)
I A general left R-module has the formM = Mm,j(Fq). Then HomR(M ,A) = Mj ,k(Fq) (viaright matrix multiplication).
I Left action of U = GL(m,Fq) on M = Mm,j(Fq):orbits O consist of row reduced echelon matrices ofsize m × j .
I Right action of Aut(A) = GL(k ,Fq) onHomR(M ,A) = Mj ,k(Fq): orbits O] consist ofcolumn reduced echelon matrices of size j × k .
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A Counter-Example to Extension (c)
I In W : F (O],Q)→ F (O,Q), the dimensions overQ of the domain and range equal the number ofelements in O] and O, respectively.
I dimQ F (O],Q) equals the number of columnreduced echelon matrices of size j × k .
I dimQ F (O,Q) equals the number of row reducedechelon matrices of size m × j .
I Since k > m, dimQ F (O],Q) > dimQ F (O,Q), andW is not injective.
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Explicit Counter-Examples (a)
I R = M1(Fq) = Fq, A = M1,2(Fq). Remember thatHamming weight depends on elements beingnonzero in A (nonzero as a pair).
I For q = 2, n = 3:
C+ C−(00, 00, 00) (00, 00, 00)(00, 10, 10) (10, 10, 00)(00, 01, 01) (00, 10, 10)(00, 11, 11) (10, 00, 10)
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Explicit Counter-Examples (b)
I For q = 3, n = 4:
C+ C−(00, 00, 00, 00) (00, 00, 00, 00)(00, 01, 01, 01) (00, 10, 20, 10)(00, 02, 02, 02) (00, 20, 10, 20)(00, 10, 10, 10) (10, 10, 10, 00)(00, 11, 11, 11) (10, 20, 00, 10)(00, 12, 12, 12) (10, 00, 20, 20)(00, 20, 20, 20) (20, 20, 20, 00)(00, 21, 21, 21) (20, 00, 10, 10)(00, 22, 22, 22) (20, 10, 00, 20)
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Characterizing Finite Frobenius Rings
I Theorem (2008). Suppose R is a finite ring, and setA = R . If the extension theorem for Hammingweight holds for linear codes over R , then R is aFrobenius ring.
I Dinh and Lopez-Permouth (2004–2005) provedsome special cases and developed a strategy toprove the general result.
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The Strategy of Dinh and Lopez-Permouth
I Every non-Frobenius ring has a copy of someMm,k(Fq) ⊂ Soc(R), with m < k .
I The extension theorem fails for Mm,k(Fq) ⊂ Soc(R),with m < k (as a module over Mm(Fq)).
I View the Mm,k(Fq) counter-examples as modules(and hence counter-examples) over R itself.
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Structure of a Finite Ring
I Let R be a finite ring with 1.
I R/Rad(R) is a sum of simple rings, which must bematrix rings over finite fields:
R/Rad(R) ∼=⊕
Mmi(Fqi
).
I Soc(RR) is a left module over R/Rad(R), so
Soc(RR) ∼=⊕
Mmi ,ki(Fqi
).
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Frobenius Rings
I Remember that a finite ring is Frobenius ifR/Rad(R) is isomorphic to Soc(R) as one-sidedmodules (so ki = mi).
I In a non-Frobenius ring, there exist ki 6= mi , withsome larger and some smaller.
I These provide the counter-examples to theextension theorem.
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Additional Comments (a)
I One can characterize alphabets A for which theextension theorem holds: A ⊂ R plus one morecondition.
I In particular, A = R always satisfies the extensiontheorem for Hamming weight (for any finite ring R ,Frobenius or not). This is a theorem of Greferath,Nechaev, Wisbauer (2004) that extends the originalFrobenius result.
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Additional Comments (b)
I Some results are known for other weight functions,especially the “homogeneous weight” (again, byGreferath, Nechaev, Wisbauer).
I But, there is much that is not known about otherweight functions. For example, it is not known if theextension theorem is always true for the Lee weightover R = Z/nZ for all n.
I Are there other uses of W : F (O],Q)→ F (O,Q)?
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References
I These slides and other papers are available on theweb: http : //homepages.wmich.edu/ ∼ jwood
I Many references in the paper “Foundations ofLinear Codes ... ”
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