On the non-existance for quantum LDPC codes of type IEEE802.16e with rates 1/2 and 2/3B

In this paper, we discuss a construction of CSS codes derived from pairs of practical irregular LDPC codes. Intersection studies between quantum error-correcting codes and LDPC codes are current hot research topics [1], [3], [4], [5], [6], [7], [8], [9]. One of aims of quantum error-correcting code theory is to construct quantum error-correcting codes with small length and high error-correcting performance. The reason why LDPC codes are chosen is their high errorcorrecting performances as classical error-correcting codes. In fact, LDPC codes have almost achieved a theoretical error-correction limit, called a Shannon limit, with various conditions, for example, code rate, code length, communicatin channel, and others. This fact induces naturally derived the quantum error-correcting codes. In the researches, we encounter a problems to hold a condition, which we call it the “twisted condition”, for a pair of classical codes to derive a quantum code. The condition requires a geometrically orderly structure to two classical code spaces. More explicitly, the dual code of one of two codes must be contained in the other code. On the other hand, a complicated structure, i.e. a random structure, is suitable for achieving high error-correcting performance. The construction theory of classical codes for a CSS code is to satisfy conflicting, orderly and complicated, requirements. We re-write the two conditions in terms of parity-check matrices. The twisted condition implies the orthogonality of their paritycheck matrices. A complex structure implies that non-zero elements in the parity-check matrices should be irregularly arranged. The previous researches on quantum LDPC codes have focused on quantum regular LDPC codes [1], [3], [7], [8] as same as that of classical LDPC codes. But now, a main stream of classical LDPC code researches is a construction method for irregular LDPC codes. The crucial reason why the main research stream has changed is error-correcting performance. In fact, the performances of irregular LDPC codes are extremely better than those of regular LDPC codes. This fact is theoretically proved by density-evolution theory and the Gaussian approximation method [10], [11]. Our design of irregular LDPC codes is based the design written in the standardization of IEEE802.16e. It is needless to say that irregular LDPC codes which are chosen in IEEE802.16e are designed well-considerably and show high error-correcting performance with practical length. Our research has tried to make a CSS code with a pair of LDPC codes of type IEEE802.16e. To our regret, we proved that it was impossible to construct a CSS code if one of classical codes was of type IEEE802.16e with rate 1/2 and 2/3B. We would like to report the discussion on its impossibility in this paper. This is the first paper to analyze the possibility of a CSS code construction by using two irregular LDPC codes which are practically useful.