The t -wise intersection and trellis of relative four-weight codes
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The t -wise intersection and trellis of relative four-weight codes B. Rega1 · Z. H. Liu2 · C. Durairajan1 Received: 1 January 2020 / Accepted: 20 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Based on the applications of codes with few weights, we define the so-called relative fourweight codes and present a method for constructing such codes by using the finite projective geometry method. Also, the t-wise intersection and the trellis of relative four-weight codes are determined. Keywords Relative four-weight code · Projective space · Support · t-wise intersecting · Trellis Mathematics Subject Classification (2010) 94B05
1 Introduction A linear code C of length n is defined as a subspace of GF(q)n , where GF(q) is a finite field with q elements, and the code C is called binary when q = 2. The t-wise intersecting codes and their wide applications were first introduced in [13], and then further studied by Cohen et al. [2, 3] and Encheva et al. [5]. The t-wise intersecting codes are a generalization of intersecting ones, which correspond to t = 2 and satisfy that any two non-zero codewords have intersecting supports. Finding the judgment criteria and a constructing method for t-wise intersecting codes is meaningful research work. The importance of another concept, the trellis of a linear code [6, 15, 17], is in that it can be used to estimate the complexity of the Viterbi decoding algorithm.
B. Rega
[email protected] Z. H. Liu [email protected] C. Durairajan [email protected] 1
Bharathidasan University, Tiruchirappalli, Tamilnadu 620 024, India
2
Beijing Institute of Technology, Beijing 100081, People’s Republic of China
Cryptography and Communications
A linear code with few weights [4] is useful in authentication codes, secret sharing schemes, and association schemes apart from its applications in consumer electronics, communication and data storage systems. Many recent papers are dedicated to constructing linear codes with few weights [7, 8, 16] by using the defining set and the technique of exponential sums. The finite geometry method was first introduced in [1] and [14], and it has been effectively generalized at present to study codes with respect to the rank-metric in [12]. By using the finite geometry method, Liu and Wu [10] provided a technique of constructing codes with few weights, namely, the so-called relative two-weight and three-weight codes. Besides the applications already mentioned for codes with few weights, Liu and Wu further showed that relative two-weight and three-weight codes can be applied to the wire-tap channel of type II with the coset coding scheme. The geometry structures of relative two-weight and three-weight codes were given in [10]. By using these geometry structures, the t-wise intersection of relative three-weight codes was calculated in [9], and the trellis of relative two-weight codes was estimated in [11]. Based on the results of relative two-weight and three-weight codes, one can find that these codes have good geo
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