Elsevier

Information Sciences

Volume 221, 1 February 2013, Pages 571-578
Information Sciences

Further results on the semilinear equivalence of linear codes

https://doi.org/10.1016/j.ins.2012.09.007Get rights and content

Abstract

Based on relative subcodes, we address an equivalent condition for a one-to-one semilinear mapping between two linear codes to be in fact a semimonomial transformation, that is, the underlying two codes are semilinearly equivalent. The result in the present paper substantially improves the equivalent condition in recent literatures. Moreover, it also generalizes the well-known MacWilliams theorem of code equivalence.

Introduction

The semilinear equivalence of codes is a generalization of a basic concept, code equivalence, in coding theory. The well-known MacWilliams theorem [12], [13] established an equivalent condition for code equivalence, which shows that any isomorphism between two linear codes C and C preserving the Hamming weight induces an equivalent relation of C and C, where “isomorphism” is a one-to-one mapping preserving linearity between two vector spaces (here linear codes are viewed as vector spaces). Although the computational complexity of the code equivalence problem is at least as hard as graph isomorphism [6], [15], [14], code equivalence and the MacWilliams theorem have yet been extensively studied because of the justification in theoretical meaning (see [1], [19], [21], [22]). Making use of the generalized Hamming weights (GHWs) [20], Fan et al. [4] established a different code equivalence, which generalized the MacWilliams theorem.

Recently, some different sufficient conditions of equivalence and semilinear equivalence of codes were introduced in [8], [9], [10] through the application of the relative generalized Hamming weights (RGHWs) [11]. In this paper, we will proceed further in this direction and address a more concise equivalent condition for semilinear equivalence of codes. The main result of the present paper substantially improves what was derived in [10] and includes the result in [8] as a special case. More concretely, the present paper generalizes the computing method used in [9] and gives answers to the common questions remaining unsolved in [8], [10], which will be explained in the later sections in detail.

The remainder of this paper is organized as follows. Section 2 sets up notation and terminology and briefly reviews some basic facts. The main results are stated and proved in Sections 3 Main results, 4 The proofs of the main results, respectively. Section 5 concludes the study.

Section snippets

Preliminaries

Let q be a power of a prime and GF(q) be a finite field with q elements, and let GF(q)n be the set consisting of all vectors with n coordinates over GF(q).

A k-dimensional linear code C with length n, usually said to be an [n,k] code, is a k-dimensional linear subspace of GF(q)n. A subcode D of C is a subspace of C. The support χ(D) of D is defined as the set of positions where not all the codewords of D have zero coordinates. In particular, the support of a codeword (or the support of the

Main results

In this section, we present an equivalent condition in Theorem 1 for a one-to-one semilinear mapping to be a semimonomial transformation. The proposed equivalent condition substantially generalizes the ones given in [9], [10]. We also present Theorem 2 as a supplement to Theorem 1. The proofs of Theorem 1, Theorem 2 will be given in Section 4.

Theorem 1

Assume that C and C have the same effective length. Let ϕ be a one-to-one semilinear mapping between C and C. Then ϕ is a semimonomial transformation if

The proofs of the main results

In this section, we denote a q-ary Gaussian binomial coefficient [17] bystq=1,t=0,(qs-1)(qs-1-1)(qs-t+1-1)(qt-1)(qt-1-1)(q-1),t0.

Based on the finite projective geometry method described in Lemma 1, we can finish the proof of Theorem 1. In a completely similar way, Theorem 2 can also be proved by using Proposition 1, and so its proof will be omitted here.

Proof of Theorem 1 To show that ϕ is a semimonomial transformation, it is sufficient to check that m(p)=m(δ(p)) for pPG(k-1,q) [[10] Lemma

Conclusion

Based on relative generalized Hamming weights, for a linear code C and its subcode C1, we first defined relative (r,s) subcodes of the pair (C,C1). We then established a tool to characterize relative subcodes, by using finite projective geometry and relative projective subspaces. Employing this tool, we proved that any one-to-one semilinear mapping between C and C of the same effective length, which preserves the support weight of each relative (r0,t0) subcode, is a semilinear equivalence of

Acknowledgements

The author would like to thank the anonymous referees for valuable remarks and helpful suggestions, and they are indebted to Chunlei Li of the University of Bergen for his help in revising this paper.

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  • Cited by (1)

    1

    The work of Z. Liu was supported by the National Natural Science Foundation of China (No. 11171366).

    2

    The work of X. Zeng was supported by the National Natural Science Foundation of China (No. 61170257).

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