The covering radii of a class of binary cyclic codes and some BCH codes

On the covering radii of a class of binary primitive cyclic codes

“…Throughout this paper, we consider binary primitive cyclic codes with n = 2 f − 1, and α denotes a primitive element of F 2 f . We begin with the following result of [9].…”

“…They also showed that if the zeros of C are α, α 2 i +1 , α 2 j +1 with distinct positive integers i, j and d(C) = 7, then R(C) = 5 for f > 8 [10, Theorem 9]. In [9], Kavut and Tutdere gave a generalization of the aforementioned results of Moreno and Castro as follows: if the zeros of C are α, α 2 i 1 +1 , . .…”

“…. , α dt , where d i ' s are distinct positive integers, and the sum of 2-weights of d i ' s, which we call ℓ, is odd such that d(C) > ℓ, then r ≤ R(C) ≤ ℓ, under some assumptions on f and r. We give a proof by generalizing the methods of [9] and [10]. This new class of cyclic codes covers that of [9], for which r = ℓ holds, and so the exact value of R(C) is obtained.…”

APN Fonksiyonlar ile Tanımlanan Bazı İkili Yarı-mükemmel Lineer Kodlar

“…In coding theory, cyclic codes are an important class of errorcorrecting codes which have favorable algebraic properties for efficient error detection and correction. In literature, there are many examples and studies on these codes, for instance see (Çalışkan, 2021), (Moreno et al, 2003), (Kavut et al, 2019). We here consider binary primitive cyclic codes defined by the APN functions over finite fields, which are linear codes.…”

“…They also showed that if the zeros of 𝐶 are 𝛼, 𝛼 2 𝑖 +1 , 𝛼 2 𝑗 +1 with distinct positive integers 𝑖, 𝑗 and 𝑑(𝐶) = 7, then 𝑅(𝐶) = 5 for 𝑚 > 8 (Moreno et al, 2003, Theorem 9). In (Kavut et al, 2019) Kavut and Tutdere gave a generalization of the aforementioned results of Moreno and Castro as follows: if the zeros of 𝐶 are 𝛼, 𝛼 2 𝑖 1 +1 , …,𝛼 2 𝑖 𝑡 +1 , where 𝑡 = (𝑟 − 1)/2, 𝑟 is any odd integer such that 𝑑(𝐶) ≥ 𝑟 + 2, then 𝑅(𝐶) = 𝑟, under some restrictions on 𝑚 and 𝑟. In (Tutdere, 2022), Tutdere proved the following: if the zeros of 𝐶 are 𝛼 𝑑 0 , 𝛼 𝑑 1 , …,𝛼 𝑑 𝑡 , where 𝑑 𝑖 's are distinct positive integers, and the sum of 2-weights of 𝑑 𝑖 's, which we call ℓ, is odd such that 𝑑(𝐶) > 𝑙, then 𝑟 ≤ 𝑅(𝐶) ≤ ℓ, under some assumptions on 𝑚 and 𝑟.…”