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Codebooks from generalized bent \(\mathbb{Z}_4\)-valued quadratic forms. (English) Zbl 1487.94200

Summary: Codebooks with small inner-product correlation have applications in unitary space-time modulations, multiple description coding over erasure channels, direct spread code division multiple access communications, compressed sensing, and coding theory. It is interesting to construct codebooks (asymptotically) achieving the Levenshtein bound. This paper presents a class of generalized bent \(\mathbb{Z}_4\)-valued quadratic forms, which contains functions proposed by Z. Heng and Q. Yue [Cryptogr. Commun. 9, No. 1, 41–53 (2017; Zbl 1380.94154)]. Using these generalized bent \(\mathbb{Z}_4\)-valued quadratic forms, we construct optimal codebooks achieving the Levenshtein bound. These codebooks have parameters \((2^{2 m} + 2^m, 2^m)\) and alphabet size 6.

MSC:

94B65 Bounds on codes
94D10 Boolean functions
11E08 Quadratic forms over local rings and fields
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
13M99 Finite commutative rings

Citations:

Zbl 1380.94154
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References:

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