Additive Cyclic Codes Funda Ozdemir Faculty of Engineering and - - PowerPoint PPT Presentation

Additive Cyclic Codes

Funda ¨ Ozdemir

Faculty of Engineering and Natural Sciences Sabancı University, Istanbul

SP Coding School, January 19-30, 2015

Funda ¨ Ozdemir Additive Cyclic Codes

Introduction

J.Bierbrauer generalized the theory of cyclic codes from the category

• f linear codes to the category of additive codes in 2002.

We will state Bierbrauer’s BCH bound on the minimum distance of additive cyclic codes. Our goal is to improve the bound on the minimum distance of these codes.

Funda ¨ Ozdemir Additive Cyclic Codes

Setting

Consider Fq with q = pe. Let n | qr − 1 and W =< α > be a multiplicative subgroup of F∗

qr .

Fix A = {i1, ..., is} ⊆ Z/nZ. Define the Fq-linear space of polynomials P(A) := {a1xi1 + · · · + asxis : a1, ..., as ∈ Fqr } Set B(A) := {(f (α0), ..., f (αn−1)) : f (x) ∈ P(A)} Define a surjective Fq-linear mapping φ : Fqr → Fm

q

x → (Tr(γ1x), ..., Tr(γmx)) for some subset {γ1, ..., γm} ⊂ Fqr , where Tr denotes the trace function from Fqr to Fq. The set {γ1, ..., γm} is linearly independent over Fq since φ is onto. Extend φ to a mapping : Fn

qr → (Fm q )n in the usual way. Funda ¨ Ozdemir Additive Cyclic Codes

Additive Cyclic Codes

Definition

Define the additive cyclic code over the alphabet Fm

q with length n as

C(A) : = (φ(B(A)))⊥ = {(φ(f (α0)), ..., φ(f (αn−1))) : f (x) ∈ P(A)}⊥ ⊆ (Fm

q )n

= {(Tr(γ1f (α0)), ..., Tr(γmf (α0)); ...; Tr(γ1f (αn−1)), ..., Tr(γmf (αn−1))) : f (x) ∈ P(A)}⊥ ⊆ Fmn

q

The code C(A) is not linear over its alphabet Fm

q . If we view C in Fmn q

as above, then it is Fq-linear. C(A) is cyclic: Since the dual code of a cyclic code is also cyclic, it is enough to show that C(A)⊥ is cyclic. Consider the codeword cf = (φ(f (α0)), ..., φ(f (αn−1))) determined by f (x) =

s

• j=1

λjxij ∈ P(A). For g(x) =

s

• j=1

λjα−ijxij ∈ P(A), we have: (φ(f (αn−1)), φ(f (α0)), ..., φ(f (αn−2))) = (φ(g(α0)), φ(g(α)), ..., φ(g(αn−1))) Linear cyclic codes correspond to the special case when m = 1 and φ(x) = Tr(x).

Funda ¨ Ozdemir Additive Cyclic Codes

BCH Bound

Definition

A ⊆ Z/nZ is an interval if there is a generator (an integer j , coprime with n) of Z/nZ such that A = {jl, j(l + 1), ..., j(l + i − 1)}, for some l (mod n). In the special case A = {i, i + 1, ..., j}, the short notation A = [i, j] is used.

Theorem (Bierbrauer’s BCH bound)

If A contains an interval of size t (mod n), then the minimum distance of C(A) is ≥ t + 1. Goal: Improve the minimum distance bound for additive cyclic codes!

Funda ¨ Ozdemir Additive Cyclic Codes