The component-wise or Schur product $C⁎C^{\prime }$ of two linear error-correcting codes C and $C^{\prime }$ over certain finite field is the linear code spanned by all component-wise products of a codeword in C with a codeword in $C^{\prime }$. When $C=C^{\prime }$, we call the product the square of C and denote it $C^{⁎2}$. Motivated by several applications of squares of linear codes in the area of cryptography, in this paper we study squares of so-called matrix-product codes, a general construction that allows to obtain new longer codes from several “constituent” codes. We show that in many cases we can relate the square of a matrix-product code to the squares and products of their constituent codes, which allow us to give bounds or even determine its minimum distance. We consider the well-known $(u,u+v)$-construction, or Plotkin sum (which is a special case of a matrix-product code) and determine which parameters we can obtain when the constituent codes are certain cyclic codes. In addition, we use the same techniques to study the squares of other matrix-product codes, for example when the defining matrix is Vandermonde (where the minimum distance is in a certain sense maximal with respect to matrix-product codes).

按组件或Schur产品 $C⁎C^{\prime }$两个线性纠错码C和$C^{\prime }$在某个有限域上，是线性代码，该线性代码由C中的代码字与C中的代码字的所有分量方式乘积跨越$C^{\prime }$。什么时候$C=C^{\prime }$，我们将乘积称为C的平方并表示它$C^{⁎2}$。基于线性码平方在密码学领域的几种应用，本文研究了所谓的矩阵乘积码的平方，该矩阵乘积码允许从多个“组成”码中获得新的较长码。我们表明，在许多情况下，我们可以将矩阵乘积代码的平方与其组成代码的平方和乘积相关联，从而使我们可以确定界限，甚至可以确定其最小距离。我们认为知名$（ü，ü+v）$-构造或Plotkin和（这是矩阵乘积码的特例），并确定当组成码是某些循环码时我们可以获得哪些参数。另外，我们使用相同的技术来研究其他矩阵乘积码的平方，例如，当定义矩阵是范德蒙德时（相对于矩阵乘积码，最小距离在某种意义上是最大）。