Characterization of completely regular codes through p-polynomial association schemes
Part of the Lecture Notes in Computer Science book series (LNCS, volume 307)
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Given a projective linear code C(n,k) with s nonzero weights we know that if C⊥ is a uniformly packed code, or more generally if C⊥ is a completely regular code, then the covering radius of C⊥ is equal to s. Furthermore, the restriction of the Hamming scheme H(n,q) to C is an association shceme and its dual association scheme is isomorphic to its Ω-scheme, an association scheme defined on the columns of the generator matrix of C.
On the other hand there exist projective linear codes C whose restriction of H(n,q) is an association scheme although its orthogonal code C⊥ is not completely regular. In other words, there exist projective linear codes C with = s but C⊥ is not completely regular.
In this paper we present two main results:
- 1.— C⊥ is a completely regular code if and only if = s and the restriction of H(n,q) on C is an association scheme.
— C⊥ is a completely regular code if and only if the Θ-scheme is a P-polynomial association scheme.
KeywordsLinear Code Association Scheme Dual Code Weight Enumerator Orthogonal Code
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