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Cyclic completely regular codes C1,5
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DOI：10.1007/s10623-026-01860-8.png)
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Abstract
En 中文
We construct a family of cyclic completely regular codes (CRCs) of length n = 2(m) - 1 and
compute their intersection arrays. These codes, denoted C-1,C-5,C- are generated by the product
m(1)(x)m(5)(x), where m(i) (x) is the minimal polynomial of alpha(i), and alpha is a primitive element of
the finite field F-2m . We consider two main cases: odd m and m equivalent to 2 (mod 4). For odd m,
these codes are known to be completely regular with covering radius rho = 3 and minimum
distance d = 5.We prove that, for any m, the codes C-1,C-3 and C-1,C-5 are non-equivalent despite
sharing the same parameters and intersection array. Form equivalent to 2 (mod 4),we demonstrate that
C-1,C-5 forms a new family of completely regular codes with covering radius rho = 3, minimum
distance d = 3, and intersection array IA = [n, n - 3, 3n+7/4; 1, 4, n-3/4]. Moreover, the
corresponding extended cyclic codes C-1,5(& lowast;) are completely regular [n+1, n-2m, 4; 4]-codes
with intersection array IA = [n + 1, n, n - 3, 3n+7/4; 1, 4, n-3/4 , n + 1]. We also describe the
parameters and some properties of the coset graphs associated to these completely regular
codes which form distance-regular graphs.
compute their intersection arrays. These codes, denoted C-1,C-5,C- are generated by the product
m(1)(x)m(5)(x), where m(i) (x) is the minimal polynomial of alpha(i), and alpha is a primitive element of
the finite field F-2m . We consider two main cases: odd m and m equivalent to 2 (mod 4). For odd m,
these codes are known to be completely regular with covering radius rho = 3 and minimum
distance d = 5.We prove that, for any m, the codes C-1,C-3 and C-1,C-5 are non-equivalent despite
sharing the same parameters and intersection array. Form equivalent to 2 (mod 4),we demonstrate that
C-1,C-5 forms a new family of completely regular codes with covering radius rho = 3, minimum
distance d = 3, and intersection array IA = [n, n - 3, 3n+7/4; 1, 4, n-3/4]. Moreover, the
corresponding extended cyclic codes C-1,5(& lowast;) are completely regular [n+1, n-2m, 4; 4]-codes
with intersection array IA = [n + 1, n, n - 3, 3n+7/4; 1, 4, n-3/4 , n + 1]. We also describe the
parameters and some properties of the coset graphs associated to these completely regular
codes which form distance-regular graphs.
Keywords:
Completely regular codes
Cyclic codes
Distance regular graphs
Extended codes
Intersection array
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