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Squares of matrix-product codes. (English) Zbl 07174346
Given two linear codes $$C$$ and $$C'$$ the Schur product is defined by $C*C'=\left\langle\{c*c'\mid c\in C, c' \in C'\}\right\rangle,$ where $$c*c'=(c_1c_1',\ldots,c_nc_n')$$.
It is well know that for some cryptographic applications, private information retrieval or multiparty computations among others, the knowledge of $$C*C'$$ is of particular interest. In certain protocols for multiparty computations, both a large minimum distance for $$C^{*2}=C*C$$ and a large dimension for $$C$$ are required. Depending on the protocol, sometimes a large minimum distance for $$C^{\perp}$$ is also demanded.
According to previous motivations, they study the structure of $$C^{*2}$$ when the code $$C$$ is a matrix product code. In the particular case of the $$(u,u+v)$$-construction, they provide a lower bound for the minimum distance which is sharp in case that the codes used in the $$(u,u+v)$$-construction are nested. Furthermore, when the constituent codes of the $$(u,u+v)$$-construction are binary cyclic codes they use the cyclotomic coset to control at the same time the dimension of $$C$$ and a lower bound of the minimum distance of $$C$$ and $$C^{*2}$$, actually they notice that considering large cyclotomic cosets one can obtain the desired codes. They are able to obtain new codes with large dimension of $$C$$ and large minimum distance of $$C^{*2}$$ simultaneously.
Finally they study matrix-product codes where the defining matrix $$A$$ is a Vandermonde matrix. Thanks to it, they can provide a better algebraic structure for $$C^{*2}$$, i.e., $$C^{*2}$$ is also a matrix-product code and a formula for the dimension and a lower bound for the minimum distance are given. In the particular case where the constituent codes of the matrix-product code are AG-codes, then more precise parameters are given.

##### MSC:
 94B05 Linear codes, general 94A62 Authentication, digital signatures and secret sharing 11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
McEliece
Full Text:
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