[Galois-disjoint] Polycyclic codes over finite chain rings LAWCI - - PowerPoint PPT Presentation

[Galois-disjoint] Polycyclic codes over finite chain rings

LAWCI 2018, Campinas

Thomas Blackford, Alexandre Fotue Tabue, Edgar Mart´ ınez-Moro

July 25th, 2018

Finite Commutative Chain Rings

A finite ring S is a finite chain ring (FCR) if S is local and principal. If θ is a generator of the maximal ideal J(S) of S the ideals of S form a chain as follows: {0S} = J(S)s J(S)s−1 · · · · · · J(S) S, and J(S)t = θtS for 0 ≤ t < s. Any FCR is the Galois extension of Zpa[θ] of degree r, for some root θ of an Eisenstein polynomial over Zpa of degree e satisfying θs−1 = θs = 0S. Whence Sr isomorphic to GR(pa, r)[θ]. We will call (pa, r, e, s) the parameters of the FCR.

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Example

Consider S = Z4[ζ][x] x2 + 2, 2x, where ζ is a root of the basic primitive polynomial f (X) = X 2 + X + 1 ∈ Z4[X]. Re has 64 elements, and is additively equivalent to Z2 ⊕ Z2 ⊕ Z4 ⊕ Z4. Moreover, it is a (Galois) extension of the ring R =

Z4[x] x2+2,2x with

S = R[ζ], and also S/x ∼ = F4 = F2(ζ). Similarly, Z4[ζ] is a Galois ring as an extension of S = Z4.

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Tree

Z4[ζ][x] x2+2,2x Z4[x] x2+2,2x

Z4[ζ] Z4 2

3 2 3 2

2

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Codes over a FCR

A S-linear code of length n is a submodule of the S-module Sn. An S-linear code over S is free, if it is free as an S-module. A matrix G is called a generator matrix for an S-linear code C, if the rows of G span C and none of them can be written as an S-linear combination

• f the other rows of G.

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Codes over a FCR

A matrix G is in the standard form if it is of the form G =     Ik0 G0,1 G0,2 · · · G0,s−1 G0,s θIk1 θG1,2 · · · θG1,s−1 θG1,s · · · · · · · · · · · · · · · · · · · · · θs−1Iks−1 θs−1Gs−1,s     U, where Ikt is an identity matrix of order kt(for 0 ≤ t < s) and U is a suitable permutation matrix. The s-tuple (k0, k1, · · · , ks−1) is called type of G.

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Polycyclic codes over a FCR

Consider the n × n-matrices Da :=      . . . In−1 a0 a1 · · · an−1      and Eb :=      b0 · · · bn−2 bn−1 In−1 . . .      with associate vectors a := (a0, a1, · · · , an−1) and b := (b0, b1, · · · , bn−1)

• f Sn, respectively.

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Polycyclic codes over a FCR

an S-linear code of length n is right polycyclic with associate vector a (or simply, right a-cyclic), if it is invariant by right multiplication of the matrix Da. It is left b-cyclic if it is invariant by right multiplication of the matrix Eb . Any right a-cyclic with a0 ∈ S× is also left b-cyclic where bj = −aj+1a−1 for j < n −1 and bn−1 = a−1

0 . Henceforth, we assume that

the right a-cyclic codes will satisfy a ∈ S× ×Sn−1, and we will simply write a-cyclic codes.

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Polycyclic codes over a FCR

Consider the S-module monomorphism Ψ : Sn ֒ → S[X] (c0, c1, · · · , cn−1) → c0 + c1X + · · · + cn−1X n−1. . C is an a-cyclic code if and only if Ψ(C) is an ideal in S[X] contained the ideal X n − Ψ( a ) in S[X] generated by X n − Ψ(a). the a-period of an a-cyclic over S will be ℓa := min

• i ∈ N\{0} : X n − π(Ψ( a )) divides X i − 1
• .

where π is the natural projection S → S/J(S). The quotient ring S[X]/ X n−Ψ( a ) is a principal ideal ring, if either S is a field or X n − π(Ψ( a )) is square free.

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Free codes

From now on (ℓa, p) = 1 A free polycyclic code C = P(S; n; g) over S is defined as follows: Consider g = g0 + g1X + · · · + gk−1X k−1 + X k over S and C is gen- erated as the spand of its cyclic shifts (as n-vectors). The polynomial g is called the generator polynomial of C. If g0 ∈ S×, and there is an a ∈ (S×) × Sn−1 such that g divides X n − Ψ(a) Then P(S; n; g) =

• c ∈ Sn : g divides Ψ( c )
• and P(S; n; g) is a-cyclic.

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Non-free codes

Every non-free, non-zero polycyclic code C over S admits a generator set [strong Gr¨

• bner basis] of the form
• θλ1g1 ; θλ2g2; · · · ; θλugu
• ∈ S[X]

such that Ψ(C) =

• θλ1g1 ; θλ2g2; · · · ; θλugu
• , and
• 1. 0 ≤ λ1 < λ2 < · · · < λu < s;
• 2. for 1 ≤ i ≤ u, gi is monic;
• 3. n > deg(g1) > deg(g2) > · · · > deg(gu) > 0;
• 4. for 0 ≤ i ≤ u, θλi+1gi ∈
• θλi+1gi+1; θλi+2gi+2; · · · ; θλugu
• .

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Non-free codes

A strong Gr¨

• bner basis is not necessarily unique.

However, u the cardinality of the basis, k1, k2, · · · , ku the degrees of its polynomials g1, g2, · · · , gu and the exponents λ1, λ2, · · · , λu are uniquely deter- mined. C =

u

• i=1

θλi (P(S; ki−1; gi))

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Euclidean orthogonality

Any S-linear code C of length n is a-cyclic, if and only if C ⊥ is a-sequential, i.e. (c1, · · · , cn−1, c, a ) ∈ C, for all c ∈ C. Sequencial codes are identified with ideals in constacyclic ambient spaces S[X]/ X n − Ψ(a) .

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Galois action

Let σ be a generator of AutSr (S). This generator σ naturally induces an automorphism of S[X] and Sn as follows: σ(f ) =

• i

σ(fi)X i, where σ( c ) = (σ(c0), σ(c1), · · · , σ(cn−1)) . Let C be an S-linear code

• 1. The σr-image of C is defined as:

σr(C) := {σr( c ) : c ∈ C} .

• 2. The code C is σr-disjoint if σir(C) ∩ C = {0} for all

1 ≤ i < d.

• 3. The code C is completed σr-disjoint if

Sn = C + σr(C) + σ2r(C) + · · · + σr(d−1)(C).

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Galois action

Let C be a non-free, non-zero a-cyclic code over S with strong Gr¨

• bner bases
• θλ1g1 ; θλ2g2; · · · ; θλugu
• .

Then σi(C) is σi(a)-cyclic, and

• θλ1σi(g1) ; θλ2σi(g2); · · · ; θλuσi(gu)
• is a Gr¨
• bner basis for σi(C), for all 0 ≤ i < m.

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Galois action

Let C := P(S; n; g) with generator polynomial g. Then C is σr- disjoint if, and only if deg(µ(g, σir(g))) ≥ n, for 1 ≤ i ≤ d − 1. Let C be an S-linear code over S of length n. If C is completed σr-disjoint then C is free.

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Trace codes and restrictions

Let C be an S-linear code of length n. The restriction code Resr(C)

• f C to Sr is defined to be Resr(C) := C ∩ (Sr)n, and the trace code

Trd(C) of C to Sr is defined as: Trd(C) :=

• (Trd(c0), Trd(c1), · · · , Trd(cn−1)) : c ∈ C
• ,

where Trd =

d−1

• i=0

σir.

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Delsarte

Let ρ ∈ S[X]. Denote by {ρ}a the unique polynomial such that X n − Ψ(a) divides ρ − {ρ}a and deg({ρ}a) < n. One defines the inner- product by x ; ya := ({Ψ(x)Ψ(y)}a) (0S). If a ∈ (S×) × Sn−1, then the bilinear form (x ; y) → x ; ya is non degenerate. The annihilator of an a-cyclic code is AnnS(C) := {y ∈ Sn : y ; ca = 0S for all c ∈ C } .

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Delsarte

AnnS(C) is also an a-cyclic code, C ⊥ and AnnS(C) have the same type, AnnS (AnnS(C)) = C and |AnnS(C)| × |C| = |S|n. Let C be a non-free, non-zero a-cyclic code over S with strong Gr¨

• bner bases
• θλ1g1 ; θλ2g2; · · · ; θλugu
• .

Let λu+1 = s and g0 = f . For i ∈ {1; 2; · · · ; u +1}, λ∗

i := s −λu−i+2

and hi a monic polynomial over S such that π (higu−i+1) := π(f ). Then AnnS(C) is an a-cyclic code with strong Gr¨

• bner basis
• θλ∗

1 h1 ; θλ∗ 2 h2; · · · ; θλ∗ u+1hu+1

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Delsarte

Let C be an a-cyclic code over S. Then Trd(AnnS(C)) = AnnSr (Resr(C)).

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Thank you for your attention!

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