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<p>We introduce a new family of rank-metric and sum-rank-metric codes called linearized Chinese Remainder Theorem codes (<i>q</i>CRT codes), constructed over linearized polynomial rings via a non-commutative generalization of the classical Chinese Remainder Theorem. After establishing the necessary algebraic foundations—including an effective CRT and its lifting for linearized polynomials—we present explicit code constructions and show that several well-known code families, such as Gabidulin or simple codes are related to <i>q</i>CRT. A probabilistic decoding algorithm is proposed for a subclass of these codes whose moduli have coefficients in a base field, with an explicit analysis of its failure rate under a uniform error model. This algorithm is further extended to a broader class of codes with moduli over small extension fields. The decoding strategy, inspired by the Chinese Remainder lifting algorithm, exploits the structure of the error support to recover the transmitted codeword. Numerical experiments illustrate the success probability as a function of the error rank weight and code parameters, highlighting the flexibility of the construction and the role of the extension degree in controlling decoding performance.</p>

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Linearized Polynomial Chinese Remainder codes

  • 期刊论文

  • 发表日期：

    2026年7月25日

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  • Philippe Gaborit,
  • Camille Garnier,
  • Olivier Ruatta

摘要

We introduce a new family of rank-metric and sum-rank-metric codes called linearized Chinese Remainder Theorem codes (qCRT codes), constructed over linearized polynomial rings via a non-commutative generalization of the classical Chinese Remainder Theorem. After establishing the necessary algebraic foundations—including an effective CRT and its lifting for linearized polynomials—we present explicit code constructions and show that several well-known code families, such as Gabidulin or simple codes are related to qCRT. A probabilistic decoding algorithm is proposed for a subclass of these codes whose moduli have coefficients in a base field, with an explicit analysis of its failure rate under a uniform error model. This algorithm is further extended to a broader class of codes with moduli over small extension fields. The decoding strategy, inspired by the Chinese Remainder lifting algorithm, exploits the structure of the error support to recover the transmitted codeword. Numerical experiments illustrate the success probability as a function of the error rank weight and code parameters, highlighting the flexibility of the construction and the role of the extension degree in controlling decoding performance.