The Hamming distances of repeated-root cyclic codes of length 5ps

Abstract

Due to the wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. In this paper, let p be a prime with p ≥ 7. We determine the weight distributions of all cyclic codes of length 5 over F q and the Hamming distances of all repeated-root cyclic codes of length 5p s over F q , where q = p m and both s and m are positive integers. Furthermore, we find all MDS cyclic codes of length 5p s and take quantum synchronizable codes from repeated-root cyclic codes of length 5p s .

Determination for minimum symbol-pair and RT weights via torsional degrees of repeated-root cyclic codes

“…From (a) and (b), we obtain the equation (6). By the same process as (a) and (b), according to n− r + k 1 > r or n− r + k 1 ≤ r, we obtain the equation ( 7) using equations ( 16) and (17). Moreover, from equation (15), we obtain all polynomials such that u-part is zero with nonzero u 2 -part:…”

“…For many years, linear codes have active developments over various rings [9,10,11,12,13,14,17,21,23]. Recently, finding symbol-pair weight and RT weight is a great challenge in coding theory [5,16,22].…”

Repeated-Root Constacyclic Codes Over the Chain Ring Fpm[u]/⟨u3⟩

“…This has encouraged many coding theorists to further study and construct MDS codes with respect to various metrics (see [20], [23], [39]). Recently, Li and Yue [24] determined Hamming distances of all repeated-root cyclic codes of length 5p s over F p m and identified all MDS codes within this class of codes, where p is a prime, s, m are positive integers and F p m is the finite field of order p m . In this paper, we shall also find MDS codes with respect to Hamming and RT metrics within the family of constacyclic codes over F p m [u]/ u 3 .…”