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arXiv:2503.13069 · Computational and Applied Mathematics

New quantum codes from homothetic-BCH codes

Carlos Galindo, Fernando Hernando, Helena Martín-Cruz

Published Dec 18, 2025• 1 citation
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Abstract

Abstract We introduce homothetic-BCH codes. These are a family of $$q^2$$ q 2 -ary classical codes $$\mathcal {C}$$ C of length $$\lambda n_1$$ λ n 1 , where $$\lambda $$ λ and $$n_1$$ n 1 are suitable positive integers such that the punctured code $$\mathcal {B}$$ B of $$\mathcal {C}$$ C in the last $$\lambda n_1 -n_1$$ λ n 1 - n 1 coordinates is a narrow-sense BCH code of length $$n_1$$ n 1 . We prove that whenever $$\mathcal {B}$$ B is Hermitian self-orthogonal, so is $$\mathcal {C}$$ C . As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths that cannot be reached by BCH codes. With this procedure, we get new quantum codes according to Grassl’s table (Grassl 2025). To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths.

DOI
10.1007/s40314-025-03570-8

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Abstract We introduce homothetic-BCH codes. These are a family of q2q^2q2 q 2 -ary classical codes C\mathcal {C}C C of length λn1\lambda n_1λn1​ λ n 1 , where λ\lambda λ λ and n1n_1n1​ n 1 are suitable positive integers such that the punctured code B\mathcal {B}B B of C\mathcal {C}C C in the last λn1−n1\lambda n_1 -n_1λn1​−n1​ λ n 1 - n 1 coordinates is a narrow-sense BCH code of length n1n_1n1​ n 1 . We prove that whenever B\mathcal {B}B B is Hermitian self-orthogonal, so is C\mathcal {C}C C . As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths that cannot be reached by BCH codes. With this procedure, we get new quantum codes according to Grassl’s table (Grassl 2025). To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths.
Abstract Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal q2q^2q2 q 2 -ary linear codes. This result can be generalized to q2mq^{2 m}q2m q 2 m -ary linear codes, m>1m > 1m>1 m > 1 . We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de , 2020) and new q-ary ones, with q≠2q \ne 2q=2 q ≠ 2 , improving others in the literature.
Three code constructions generating new families of good nonbinary quantum codes are presented in this paper. The first two ones are derived from Hermitian self-orthogonal non-narrow-sense Bose-Chaudhuri-Hocquenghem (BCH) codes. The third one is derived from qqq-ary ($q\ensuremath{\ne}2$ is a prime power) Steane's enlargement of Calderbank-Shor-Steane codes applied to Euclidean self-orthogonal non-narrow-sense BCH codes. The quantum nonbinary BCH codes presented here have parameters better than the ones available in the literature.
The Bose–Chaudhuri–Hocquenghem (BCH) codes have been studied for more than 57 years and have found wide application in classical communication system and quantum information theory. In this paper, we study the construction of quantum codes from a family of [Formula: see text]-ary BCH codes with length [Formula: see text] (also called antiprimitive BCH codes in the literature), where [Formula: see text] is a power of 2 and [Formula: see text]. By a detailed analysis of some useful properties about [Formula: see text]-ary cyclotomic cosets modulo [Formula: see text], Hermitian dual-containing conditions for a family of non-narrow-sense antiprimitive BCH codes are presented, which are similar to those of [Formula: see text]-ary primitive BCH codes. Consequently, via Hermitian Construction, a family of new quantum codes can be derived from these dual-containing BCH codes. Some of these new antiprimitive quantum BCH codes are comparable with those derived from primitive BCH codes.
In this paper, we study a family of constacyclic BCH codes over Fq2\mathbb{F}_{q^2}Fq2​ of length n=q2m−1q+1n=\frac{q^{2m}-1}{q+1}n=q+1q2m−1​, where qqq is a prime power, and m≥2m\geq2m≥2 an even integer. The maximum design distance of narrow-sense Hermitian dual-containing constacyclic BCH codes of length nnn is determined. Furthermore, the exact dimension of the constacyclic BCH codes with given design distance is computed. As a consequence, we are able to derive the parameters of quantum codes as a function of their design parameters of the associated constacyclic BCH codes. This improves the result by Yuan et al. (Des Codes Cryptogr 85(1): 179-190, 2017), showing that with the same lengths, except for three trivial cases (q=2,3,4q=2,3,4q=2,3,4), our resultant quantum codes can always yield strict dimension or minimum distance gains than the ones obtained by Yuan et al.. Moreover, fixing length n=q2m−1q+1n=\frac{q^{2m}-1}{q+1}n=q+1q2m−1​, some constructed quantum codes have better parameters or are beneficial complements compared with some known results (Aly et al., IEEE Trans Inf Theory 53(3): 1183-1188, 2007, Li et al., Quantum Inf Process 18(5): 127, 2019, Wang et al., Quantum Inf Process 18(8): 323, 2019, Song et al., Quantum Inf Process 17(10): 1-24, 2018.).
Classical Bose–Chaudhuri–Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of lengthnnncan contain its dual code only if its designed distanceδ=O(n)\delta =O(\sqrt {n})δ=O(n​), and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and – consequently – the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters.
Let q≥3q\geq 3q≥3 be a prime power. Maximal designed distances of primitive Hermitian dual containing q2q^{2}q2-ary BCH codes (narrow-sense or non-narrow-sense) are determined by a careful analysis of properties of cyclotomic cosets. Non-narrow-sense BCH codes which achieve these maximal designed distances are presented, and a sequence of nested non-narrow-sense BCH codes that contain these BCH codes with maximal designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these non-narrow-sense BCH codes. The nonbinary quantum BCH codes presented here have better parameters than those quantum BCH codes available in the literature.