On codes over the finite non chain ring A = F4+vF4; v2 = v and its covering radius of codes with Bachoc weight

Abstract

In this paper, some lower and upper bounds on the covering radius of codes over the nite non chain ring A = F4 + vF4; v2 = v with respect to Bachoc weight is given. Also, the covering radius of various Block Repetition Codes of same and different length over the nite non chain ring A = F4 + vF4; v2 = v is obtained.In this paper, some lower and upper bounds on the covering radius of codes over the nite non chain ring A = F4 + vF4; v2 = v with respect to Bachoc weight is given. Also, the covering radius of various Block Repetition Codes of same and different length over the nite non chain ring A = F4 + vF4; v2 = v is obtained.

On DNA code over a finite ring and its related parameters

“…There are many researchers doing research on code over finite rings. In particular, codes over Z 4 received much attention [2,3,4,9,11,15,16,5]. The covering radius of binary linear codes were studied [4,5].…”

“…In particular, codes over Z 4 received much attention [2,3,4,9,11,15,16,5]. The covering radius of binary linear codes were studied [4,5]. Recently the covering radius of codes over Z 4 has been investigated with various distances [12].…”

“…In 1999, Sole et al gave many upper and lower bounds on the covering radius of a code over Z 4 with different distances. In [14,5], the covering radius of some particular codes over Z 4 have been investigated.…”

“…r M α k,t ≤ 5 3 (4 k − 4 r ) + r M α r,t for t < r ≤ k.Proof. In equation(5.1), Proposition 3.1, Theorem 3.3 and[5], thusGG • • • G >) + r M α r,t = 5.4 k−1 + r M α k−1,t , for k ≥ r > t. ≤ 5.4 k−1 + 5.4 k−2 + • • • + 5.4 r + r M α r,t for k ≥ r > t r M α k,t ≤ 5 − 4 r ) + r M α r,t , fork ≥ r > t. Theorem 5.2. r M β k,t ≤ 2 k (5•2 k −6)+2 r (6−5•2 r ) r M β r,t , for t < r ≤ k.Proof. Using Proposition 3.1, Theorem 3.5, in equation(5.2) and[5], obtain (5 • 2 k − 6) + 2 r (6 − 5 • 2 r ) 6 + r M β r,t , for t < r ≤ k.…”