Abstract We investigate unbiased weighing matrices of weight 9 and provide a construction method using mutually suitable Latin squares. For $$n \le 16$$ n ≤ 16 , we determine the maximum size among sets of mutually unbiased weighing matrices of order n and weight 9. Notably, our findings reveal that 13 is the smallest order where such pairs exist, and 16 is the first order for which a maximum class of unbiased weighing matrices is found.

In this paper, we explore the application of semidef-inite programming to the realm of quantum codes, specifically focusing on codeword stabilized (CWS) codes with entanglement assistance. Notably, we utilize the isotropic subgroup of the CWS group and the set of word operators of a CWS-type quantum code to derive an upper bound on the minimum distance. Furthermore, this characterization can be incorporated into the associated distance enumerators, enabling us to construct semidefinite constraints that lead to SDP bounds on the minimum distance or size of CWS-type quantum codes. We illustrate several instances where SDP bounds outperform LP bounds.

In this paper, we explore the application of semidefinite programming to the realm of quantum codes, specifically focusing on codeword stabilized (CWS) codes with entanglement assistance. Notably, we utilize the isotropic subgroup of the CWS group and the set of word operators of a CWS-type quantum code to derive an upper bound on the minimum distance. Furthermore, this characterization can be incorporated into the associated distance enumerators, enabling us to construct semidefinite constraints that lead to SDP bounds on the minimum distance or size of CWS-type quantum codes. We illustrate several instances where SDP bounds outperform LP bounds, and there are even cases where LP fails to yield meaningful results, while SDP consistently provides tighter and relevant bounds. Finally, we also provide interpretations of the Shor-Laflamme weight enumerators and shadow enumerators for codeword stabilized codes, enhancing our understanding of quantum codes.

We study the maximum size of a binary code A(n, d) with code length n and minimum distance d. Schrijver studied the Terwilliger algebra of the Hamming scheme and proposed a semidefinite program to upper bound A(n, d). We derive additional semidefinite constraints based on a split Terwilliger algebra so that Schrijver’s semidefinite programming bounds on A(n, d) can be improved. In particular, we show that A(18, 4) ≤ 6551 and A(19, 4) 13087.

A set of lines in $\mathbb{R}^n$ is called equiangular if the angle between each pair of lines is the same. We derive new upper bounds on the cardinality of equiangular lines. Let us denote the maximum cardinality of equiangular lines in $\mathbb{R}^n$ with the common angle $\arccos \alpha$ by $M_{\alpha}(n)$. We prove that $M_{\frac 1 a} (n) \leq \frac 1 2 ( a^2-2) ( a^2-1)$ for any $n \in \mathbb{N}$ in the interval $a^2 -2 \leq n \leq 3 a^2-16$ and $a \geq 3$. Moreover, we discuss the relation between equiangular lines and spherical two-distance sets and we obtain the new results on the maximum spherical two-distance sets in $\mathbb{R}^n$ up to $n \leq 417$.