Junmin An

Junmin An, Jae-Hyun Baek, Keon-Hwi Kim et al.

This paper presents a novel construction method for symmetric Sudoku-type games based on Lee distance perfect codes and diameter perfect codes. The proposed method utilizes the tiling property of these codes to define the structure of the subgrid constraints of Sudoku-type games. In this way, our games inherit the symmetric properties of Sudoku. We provide a detailed analysis of two small cases: a $5 \times 5$ Sudoku in $\mathbb{Z}_5^2$, and an $8 \times 8$ Sudoku in $\mathbb{Z}_8^2$. By defining equivalence relations via rigid motions, we provide a complete enumeration of valid grids, identifying 17 inequivalent solutions for $5\times 5$ Sudoku. For two different types of $8\times 8$ Sudoku, we characterize 232,735 and 304,014 inequivalent solutions, respectively. Furthermore, to verify practical playability, we implement a human-like solver that assesses the difficulty of the generated games. The analysis confirms that our $5\times5$ Sudoku games offer a balanced distribution of difficulty levels, ranging from Easy to Hard, making them a viable alternative to traditional $9 \times 9$ Sudoku.

Minjia Shi, Xuan Wang, Junmin An et al.

We study linear codes over Gaussian integers equipped with the Mannheim distance. We develop Mannheim-metric analogues of several classical bounds. We derive an explicit formula for the volume of Mannheim balls, which yields a sphere packing bound and constraints on the parameters of two-error-correcting perfect codes. We prove several other useful bounds, and exhibit families of codes meeting these bounds for some parameters, thereby showing that these bounds are tight. We also discuss self-dual codes over Gaussian integers and obtain upper bounds on their minimum Mannheim distance for certain parameter regions using a Mannheim version of the Macwilliams-type identity. Finally, we present decoding algorithms for codes over Gaussian integer residue rings. We give examples showing that certain errors which are not correctable under the Hamming metric become correctable under the Mannheim metric.