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

JW (WMU) Finite Frobenius Rings November 7, 2011 2 / 40

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?


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.


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).


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?


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)


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.


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.)


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.


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.


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).


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.


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).


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.


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 .


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 .


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 .


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.


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.


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).


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.


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.).


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.


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 .


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 .


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 .


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 .


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 .


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 .


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.


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)


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)


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.


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.


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

).


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.


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.


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)?


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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Characterizing Finite Frobenius Rings Via Coding Theory