Optimal binary and ternary linear codes with hull dimension one
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Optimal binary and ternary linear codes with hull dimension one Todsapol Mankean1 · Somphong Jitman1 Received: 13 February 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract Hulls of linear codes have been of interest and extensively studied due to their wide applications. In this paper, we focus on constructions and optimality of linear codes with hull dimension one over small finite fields. General constructions for such codes are given together with the analysis on their parameters. Optimal linear [n, 2, d]q codes with hull dimension one are presented for all positive integers n ≥ 3 and q ∈ {2, 3}. Moreover, for q = 2, the enumeration of such optimal codes is given up to equivalence. Keywords Linear codes · Optimal linear codes · Griesmer bound · Euclidean inner product · Hermitian inner product · Hulls · Hull dimensions Mathematics Subject Classification 94B05 · 94B60
1 Introduction The notion of hulls has been first introduced and applied in the characterization of finite projective planes in [1]. Hulls of linear codes have been extensively studied due to their wide applications. The complexity of some algorithms in coding theory has been determined by the hull dimension of codes [13,14,22,24]. The enumeration of linear codes with common hull dimension and the average hull dimension of linear codes have been studied in [21]. Recently, rigorous treatment for the hulls of linear
This research was supported by the Thailand Research Fund and Silpakorn University under Research Grant RSA6280042.
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Somphong Jitman [email protected] Todsapol Mankean [email protected]
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Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand
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T. Mankean, S. Jitman
codes has been given and applied in constructions of entanglement-assisted quantum error correcting codes with good parameters in [8,15,28]. Some families of linear codes with special hulls such as self-orthogonal codes and linear complementary dual (LCD) codes have practical applications in communications systems and link with other objects as shown in [2–6,8,10–12,16,18,19,23,25,26] and references therein. Precisely, self-orthogonal codes are linear codes with maximal hull and LCD codes are linear codes of minimal hull. In [16], it has been shown that asymptotically good LCD codes exist. In [23], it has been shown that LCD codes meet the asymptotic Gilbert-Varshamov bound. Optimal and good LCD codes have been studied in [4,7,10,17]. Bounds and optimality of self-orthogonal codes have been given in [11,18,20,27]. Based on the discussion above, it is therefore of interest to study constructions and optimality of linear codes with prescribed hull dimension and their applications. As mentioned in [13,14,22,24], linear codes with small hull can reduce the complexity of some algorithms in coding theory. Therefore, it is of interest to study linear codes with small hull dimension. Here, we extend techniques used in [17] to construct optimal linear codes with dimension two and hu
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