Properties of constacyclic codes under the Schur product

Abstract

For a subspace W of a vector space V of dimension n, the Schur-product space \(W^{\left\langle k \right\rangle }\) for \(k \in {\mathbb {N}}\) is defined to be the span of all vectors formed by the component-wise multiplication of k vectors in W. It is well known that repeated applications of the Schur product to the subspace W creates subspaces \(W, W^{\left\langle 2 \right\rangle }, W^{\left\langle 3 \right\rangle }, \ldots \) whose dimensions are monotonically non-decreasing. However, quantifying the structure and growth of such spaces remains an important open problem with applications to cryptography and coding theory. This paper characterizes how increasing powers of constacyclic codes grow under the Schur product and gives necessary and sufficient criteria for when powers of the code and/or the dimension of the code are invariant under the Schur product.

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Acknowledgements

This work was supported by the National Science Foundation under grants no. CNS-1651344 and CNS-1513671. Heninger and Rudow carried out this research while at the University of Pennsylvania.

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Correspondence to Michael Rudow.

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Falk, B.H., Heninger, N. & Rudow, M. Properties of constacyclic codes under the Schur product. Des. Codes Cryptogr. 88, 993–1021 (2020). https://doi.org/10.1007/s10623-020-00720-3

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Keywords

  • Constacyclic
  • Schur product
  • Codes

Mathematics Subject Classification

  • 94B15
  • 94B05