In this paper, a construction of a pair of "regular" quasi-cyclic LDPC codes to construct a quantum errorcorrecting code is proposed. In other words, we find quantum regular LDPC codes with various weight distributions. Our construction method is based on algebraic combinatorics and achieves a lower bound of the code length, and has lots of variations for length, code rate. These codes are obtained by a descrete mathematical characterization for model matrices of quasi-cyclic LDPc codes.I. INTRODUCTION Quantum error-correcting codes provide detection or correction of the errors which occur in a communication through a noisy quantum channel. Hence the codes protect integrity of a message sent from a sender to a receiver. This paper presents a construction method of a pair of "regular" quasi-cyclic low-density parity-check codes (regular QC-LDPC codes) as ingredients of a CSS code. QC-LDPC codes are known as a practical class of classical error-correcting codes due to its compact representation and good performance, especially for short code lengths [1], [2]. Since CSS codes find their applications not only for quantum error-correction but also for privacy amplification of quantum cryptography, they have become important research objects for quantum information theory [3].Mackay proposed bicycle codes which are constructed by a combination of heuristic method and theoretical approach [4]. In his paper, we find the following requirement "we delete rows using the heuristic that the column weight of a matrix should be as uniform as possible". This implies that the design of weight distribution for quantum LDPC codes by deterministic method is a theoretically interesting problem. Recently, various construction methods of quantum LDPC codes have been proposed [5], [6], [7], [8], [9], [10]. Among these proposed constructions, it has been difficult to design the weight distribution of the related parity-check matrices. In this paper, we give a solution for the weight distribution design problem for regular weight case.Contributions of this paper are the following: 1. We find a characterization for model matrices of QC-LDPC codes with circulant permutation matrices to be ingredient codes of a CSS code. The characterization is easily treatable for not only theoretical use but also computer experiments. 2. We propose a construction for (λ 1 , ρ)-regular and (λ 2 , ρ)-regular parity-check matrices for any 1 ≤ λ 1 , λ 2 ≤ ρ/2 such that the associated QC-LDPC codes are ingredient codes of a CSS code. In other words, we find quantum regular LDPC codes. Our method for designing a pair of LDPC matrices is theoretical and deterministic.

It has been proven that the code lengths of Tardos's collusion-secure fingerprinting codes are of theoretically minimal order with respect to the number of adversarial users (pirates). However, the code lengths can be further reduced as some preceding studies have revealed. In this article we improve a recent discrete variant of Tardos's codes, and give a security proof of our codes under an assumption weaker than the original Marking Assumption. Our analysis shows that our codes have significantly shorter lengths than Tardos's codes. For example, when c = 8, our code length is about 4.94% of Tardos's code in a practical setting and about 4.62% in a certain limit case. Our code lengths for large c are asymptotically about 5.35% of Tardos's codes.Communicated by H. van Tilborg.A part of this work was presented at 17th Applied Algebra, Algebraic Algorithms, and Error Correcting Codes (AAECC-17),

For designing low-density parity-check (LDPC) codes for quantum error-correction, we desire to satisfy the conflicting requirements below simultaneously. 1) The row weights of parity-check "should be large": The minimum distances are bounded above by the minimum row weights of parity-check matrices of constituent classical codes. Small minimum distance tends to result in poor decoding performance at the error-floor region.2) The row weights of parity-check matrices "should not be large": The performance of the sum-product decoding algorithm at the water-fall region is degraded as the row weight increases. Recently, Kudekar et al. showed spatially-coupled (SC) LDPC codes exhibit capacity-achieving performance for classical channels. SC LDPC codes have both large row weight and capacity-achieving error-floor and water-fall performance. In this paper, we propose a new class of quantum LDPC codes based on spatially coupled quasi-cyclic LDPC codes. The performance outperforms that of quantum "non-coupled" quasi-cyclic LDPC codes.

A set of linearly constrained permutation matrices are proposed for constructing a class of permutation codes. Making use of linear constraints imposed on the permutation matrices, we can formulate a minimum Euclidian distance decoding problem for the proposed class of permutation codes as a linear programming (LP) problem. The main feature of this novel class of permutation codes, called LP decodable permutation codes, is this LP decodability. It is demonstrated that the LP decoding performance of the proposed class of permutation codes is characterized by the vertices of the code polytope of the code. In addition, based on a probabilistic method, several theoretical results for randomly constrained permutation codes are derived.