On the dimension of twisted centralizer codes

Abstract

Given a field F , a scalar λ ∈ F and a matrix A ∈ F n×n , the twisted central-where c A (t) denotes the characteristic polynomial of A. We also show how C F (A, λ) decomposes, and we estimate the probability that C F (A, λ) is nonzero when |F | is finite. Finally, we prove dim C F (A, λ) n 2 /2 for λ ∈ {0, 1} and 'almost all' n × n matrices A over F .

Twisted centralizer codes

“…In the present paper, we extend and sometimes correct the results of [1]. In particular the incorrect [1, Theorem 2.4] is corrected and generalized for this larger class of codes in [2] (see Theorem 2.3), and we exploit this result in several ways in Section 2.2.…”

“…First assume that i 1 = i 2 and j 1 = j 2 , and let (2). As each of the diagonal entries of this restriction is nonzero, and as S = S ′ + b 3 S(i 3 , j 3 ) = 0, it follows that (i) either i 3 = i 1 or j 3 = j 1 ; and also (ii) either i 3 = i 2 or j 3 = j 2 .…”

“…The ternary codes with parameters [9,5,4], [9,3,6], [9,2,6], [9,1,9] are all optimal ternary codes according to [6]. This contrasts with ordinary centralizer codes, where the minimum distances are at most n (and here n = 3).…”

“…If A = 1 1 4 4 ∈ F 2×2 5 , then C(A, 2) is an F 5 -linear code with parameters [4,2,3]. Note that this code is also an MDS-code.…”

“…Recall that it is impossible to have a one-error correcting code from the centralizer of a 2 × 2 matrix, but we are able to do so with the 2-twist. give optimal codes C(A 1 , 2), C(A 2 , 2), C(A 3 , 2) with parameters [9,2,7], [9,3,6] and [9,5,4], respectively. …”