On dual-containing, almost dual-containing matrix-product codes and related quantum codes

Matrix-product (MP) codes are a type of long codes formed by combining several commensurate constituent codes with a defining matrix. In this paper, we study the MP code when the defining matrix A satisfies the condition that $A{A}^{\top }$ is $(D,\tau )$-monomial. We give an explicit formula for calculating the dimension of the hull of a MP code. We present the necessary and sufficient conditions for a MP code to be dual-containing (DC), almost dual-containing (ADC), self-orthogonal (SO) and almost self-orthogonal (ASO), respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code that is DC, ADC, SO and ASO, respectively. We give alternative necessary and sufficient conditions for a MP code to be ADC and ASO, respectively, and show several cases where a MP code is not ADC or ASO. We give the construction methods of DC and ADC MP codes, including those with optimal minimum distance lower bounds. We introduce the notation of τ-optimal defining (τ-OD) matrices and provide the criteria for determining whether two types of $k\times k$ matrices are τ-OD matrices at $k=3$ and $k=4$, respectively. We give many examples of DC and ADC MP codes involving τ-OD matrices, some of which are optimal or almost optimal according to the Database [11]. By applying the generalized Steane's enlargement procedure to these DC MP codes, we obtain some good quantum codes that improve those available in the Database [7].