Type II codes over F2m+uF2m

In this work, we investigate linear codes over the ring F 2 + uF 2 + vF 2 + uvF 2 . We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43-65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32-45, 1999). We then characterize the F 2 + uF 2 + vF 2 + uvF 2 -linearity of binary codes under the Gray map and give a main class of binary codes as an example of F 2 + uF 2 + vF 2 + uvF 2 -linear codes. The duals and the complete weight enumerators for F 2 + uF 2 + vF 2 + uvF 2 -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over F 2 + uF 2 + vF 2 + uvF 2 are obtained. Mathematics Subject Classifications (2000)94B05 · 94B99 · 11T71 · 13M99 IntroductionLinear codes over rings have been studied extensively in the past decade, with the emergence of the ground-breaking work done in [5] in which they looked at linear codes over Z 4 . Since then, many different types of rings have been studied in connection with coding Communicated by

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“…It is easy to see that the generators are independent in our sense and hence the code has type (16) 1 …”

Section: Lemma 27 Let C Be a Linear Code Over The Ringmentioning

confidence: 98%

“…3, an extension of the Lee weight defined for F 2 + uF 2 -codes done in works like [1,3,4], etc. will be introduced for linear codes over F 2 + uF 2 + vF 2 + uvF 2 together with a Gray map that is a weight preserving isometry.…”

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confidence: 99%

Linear codes over $${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$$

Yıldız

Karadeniz

2009

Des. Codes Cryptogr.

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“…Linear codes over the ring R m = F 2 m +uF 2 m were studied [1]. In that work, the authors used the notion of a Trace Orthogonal Basis to define the Lee weight on this ring.…”

Section: The Ring F 2 M + Uf 2 M and The Lee Weightmentioning

confidence: 99%

“…In the same work, the authors proved that ψ m,1 is a Lee weight preserving map with the Lee weight for R 1 = F 2 +uF 2 defined as in the previous section. We can summarize them in the following two lemmas which will be useful for us later: 1 …”

Section: Definition 35 a Linear Code Over The Ring R M = F 2 M + Uf mentioning

confidence: 99%

Weights modulo p e of linear codes over rings

Yıldız

2007

Des Codes Crypt

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In this paper we look at linear codes over the Galois ring GR(p , m) with the homogeneous weight and we prove that the number of codewords with homogeneous weights in a particular residue class modulo p e are divisible by high powers of p. We also state a result for a more generalized weight on linear codes over Galois rings. We obtain similar results for the Lee weights of linear codes over F 2 m + uF 2 m and we prove that the results we obtain are best possible. The results that we obtain are an improvement to Wilson's results in [Wilson RM (2003) In: Proceedings of international workshop on Cambridge linear algebra and graph coloring]

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“…A self-dual code over F 2 r + uF 2 r is of Type II if and only if the Lee weight of every codeword is a multiple of 4 [3]. Let x = (x 1 , .…”

Section: Type II Codesmentioning

confidence: 99%

On the Algebraic Structure of Quasi-cyclic Codes IV: Repeated Roots

2006

Self Cite

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A trace formula for quasi-cyclic codes over rings of characteristic not coprime with the co-index is derived. The main working tool is the Generalized Discrete Fourier Transform (GDFT), which in turn relies on the Hasse derivative of polynomials. A characterization of Type II self-dual quasi-cyclic codes of singly even co-index over finite fields of even characteristic follows. Implications for generator theory are shown. Explicit expressions for the combinatorial duocubic, duoquintic and duoseptic constructions in characteristic two over finite fields are given.

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Type II codes over F2m+uF2m

Cited by 28 publications

References 7 publications

Linear codes over $${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$$

Linear codes over $${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$$

Weights modulo p e of linear codes over rings

On the Algebraic Structure of Quasi-cyclic Codes IV: Repeated Roots

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