Tue. May 28th, 2024

Lement of R. The -constacyclic shift on Rn is defined as (( x0 , x1 , . . . , xn-1)) = (xn-1 , x0 , x1 , . . . , xn-2), in addition to a code C is said to become -constacyclic if (C) = C, i.e., if C is closed beneath the constacyclic shift . In case = 1, these -constacyclic codes are called cyclic codes, and when = -1, such -constacyclic codes are known as negacyclic codes. Every codeword c = (c0 , c1 , . . . , cn-1) C is Mitapivat Activator customarily identified with its polynomial representation c( x) = c0 c1 x cn-1 x n-1 , and the code C is in turn identified with all the set of all polynomial representations of its codewords. Then inside the ring R[ x ]/ x n – , xc( x) corresponds to a -constacyclic shift of c( x). From that, the following truth follows at when (cf. [28,29]). Proposition 1. A linear code C of length n is -constacyclic over R if and only if C is an best of R[ x ]/ x n – . Let be an alphabet of size q, whose elements are called symbols. Suppose that x = ( x0 , x1 , . . . , xn-1) can be a vector in n . The symbol-pair vector of x is defined as ( x) = (( x0 , x1), ( x1 , x2), . . . , ( xn-1 , x0)). In 2010, Cassuto and Blaum [1] gave the definition in the symbol-pair distance as the Hamming distance over the alphabet (,). Provided x = ( x0 , x1 , . . . , xn-1), y = (y0 , y1 , . . . , yn-1), the symbol-pair distance involving x and y is defined as dsp ( x, y) = d H ( ( x), (y)) = |i |. The symbol-pair distance of a symbol-pair code C is defined as dsp (C) = min x, y C, x = y. The symbol-pair weight of a vector x is defined because the Hamming weight of its symbolpair vector ( x): wtsp ( x) = |i |. In the event the code C is linear, its symbol-pair distance is equal to the Birinapant Epigenetics minimum symbol-pair weight of nonzero codewords of C: dsp (C) = min 0 = x C .Mathematics 2021, 9,3 ofThroughout this paper, let p be a prime, s, m be optimistic integers, F pm be the finite field of order pm , and let R = F pm [u]/ u3 be the finite commutative chain ring with unity. By applying Proposition 1, all -constacyclic codes of length ps over R are precisely the ideals within the ring s R = R[ x ]/ x p – , where can be a nonzero element of F pm . In [26], Laaouine et al. classified all -constacyclic codes of length ps over R. Theorem 1 (cf. [26]). The ring R is often a nearby finite non chain ring with maximal best u, x – 0 , exactly where 0 F pm such that 0 = . The -constacyclic codes of length ps more than R, that may be, ideals from the ring R , are Sort 1 (C1) : 0 , Variety two (C2) : Form 3 (C3) : 1 .psC2 = u2 ( x – 0) , where 0 ps – 1. C3 = u( x – 0) u2 ( x – 0)t h( x) ,exactly where 0 L ps – 1, 0 t L, either h( x) is 0 or h( x) is often a unit in R . Here L would be the smallest integer such that u2 ( x – 0)L C3 .Form 4 (C4) :C4 = u( x – 0) u2 ( x – 0)t h( x), u2 ( x – 0) ,exactly where 0 L ps – 1, 0 t , either h( x) is actually a unit in R or 0, and L is exact same as in Kind 3.Sort 5 (C5) :C5 = ( x – 0) a u( x – 0)t1 h1 ( x) u2 ( x – 0)t2 h2 ( x) ,where 0 V U a ps – 1, 0 t1 U, 0 t2 V, either h1 ( x), h2 ( x) are 0 or are units in R . Here U getting the smallest integer such that u( x – 0)U u2 g( x) C5 , for some g( x) R and V will be the smallest integer satisfying u2 ( x – 0)V C5 .Kind six (C6) :C6 = ( x – 0) a u( x – 0)t1 h1 ( x) u2 ( x – 0)t2 h2 ( x), u2 ( x – 0)c ,where 0 c V U a ps – 1, 0 t1 U, 0 t2 c, either h1 ( x), h2 ( x) are 0 or are units in R . Here U, V are exact same as in Variety 5.Form 7 (C7) :C7 = ( x – 0) a u( x – 0)t1 h1 ( x) u2 ( x – 0)t2 h2 ( x), u( x – 0)b u2 ( x – 0).