On some conjectures about optimal ternary cyclic codes
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On some conjectures about optimal ternary cyclic codes Qian Liu1,2 · Ximeng Liu1,2 Received: 29 May 2020 / Revised: 3 August 2020 / Accepted: 29 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, by investigating the solutions of certain equations over finite fields, we make progress towards three conjectures about optimal ternary cyclic codes which were proposed by Ding and Helleseth. Keywords Cyclic code · Optimal code · Ternary code · Sphere packing bound
1 Introduction Cyclic codes are an important subclass of linear codes and have been extensively studied [9]. Let p be a prime and m be a positive integer. Let 𝔽p and 𝔽pm denote the finite fields with p and pm elements, respectively. An [n, k, d] linear code C over the finite field 𝔽p is a k-dimensional subspace of 𝔽pn with minimum Hamming distance d, and is called cyclic if any cyclic shift of a codeword is another codeword of C . Assume gcd(n, p) = 1 . By identifying any vector (c0 , c1 , ⋯ , cn−1 ) ∈ 𝔽pn with
c0 + c1 x + c2 x2 + ⋯ + cn−1 xn−1 ∈ 𝔽p [x]∕(xn − 1), any cyclic code of length n over 𝔽p corresponds to an ideal of the polynomial residue class ring 𝔽p [x]∕(xn − 1) . It is well known that every ideal of 𝔽p [x]∕(xn − 1) is principal. The cyclic code can be expressed as C = ⟨g(x)⟩ , where g(x) is monic and has the least degree among all elements in C . Then g(x) is called the generator polynomial of C and h(x) = (xn − 1)∕g(x) is referred to as the parity-check polynomial of C . Due to their wide applications in mathematics and engineering, such as cryptography [1] and sequence design [4], much progress had been made on cyclic codes in the past * Qian Liu [email protected] 1
College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China
2
Key Lab of Information Security of Network Systems, Fuzhou University, Fuzhou 350108, China
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few decades. For some advances in cyclic codes we refer to [2, 6, 7, 10, 16, 18, 19, 21] and the references therein. Let 𝛼 be a generator of 𝔽3∗m = 𝔽3m �{0} and mi (x) be the minimal polynomial of 𝛼 i over 𝔽3 , where 1 ≤ i ≤ 3m − 1 . Let u, v be two integers such that 𝛼 u is not a Galois conjugate of 𝛼 v , the cyclic code over 𝔽3 with generator polynomial mu (x)mv (x) is denoted by C(u,v) . In recent years, many scholars were interested in studying optimal cyclic codes over finite fields with respect to the Sphere packing bound [9]. For (u, v) = ((3m + 1)∕2, (3k + 1)∕2) , where m is odd and k is even, Zhou and Ding [22] gave a class of optimal ternary cyclic codes with parameters [3m − 1, 3m − 1 − 2m, 4] . In 2016, for (u, v) = ((3m + 1)∕2, 2 ⋅ 3(m−1)∕2 + 1) , where m ≥ 3 is odd and v is a Welch-type exponent, Fan et al. [5] obtained a new class of optimal ternary cyclic codes and discussed the duals of them. After th
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