Binary Construction of Quantum Codes of Minimum Distance Three and Four

R. Li, X. Li
2004 IEEE Transactions on Information Theory
1331 on periodic sequences. In particular, for case i), note that the condition is violated because for any d 1, the shift space S(d; 2d) contains a sequence of period d + 1, namely, the sequence (0 d 1) 1 = 11 10 d 1 0 d 1 0 d 1 . . ., while S(d + 1; 3d + 1) does not contain any sequence of period d + 1 or less. The periodic sequence condition is violated in case ii) as well, since S(d; 1) contains the all-zeros sequence 0 1 , which is periodic with period 1, while S(d 0 1; 2d 0 1), d > 1,
more » ... not contain the all-zeros sequence or the all-ones sequence, which are the only sequences of period 1. We would like to remark that when d = 1, the shift space S(d01; 2d 01) does in fact contain the all-ones sequence. The condition on periodic sequences is not strong enough to handle case iii), since S(4; 1) contains the period-1 sequence consisting of all zeros. Instead, we show that there cannot exist a rate 1:1 slidingblock decodable encoder from S(4; 1) to either S(1; 2) or S(2; 4) by appealing to the characteristic polynomial condition. From (3) , we see that 4;1 (z) = z 5 0 z 4 0 1 and from (2) we find that 1;2 (z) = z 3 0 z 0 1 and 2;4 (z) = z 5 0 z 2 0 z 0 1: By inspection, it follows that 4;1 (z) is not a divisor of either 1;2 (z) or 2;4(z) in the ring of integer polynomials. Therefore, there does not exist a sliding-block mapping from either S(1; 2) or S(2; 4) onto S(4; 1). This completes the proof of Theorem I.1. We would like to make one final observation regarding the (d; k)-constrained shift spaces S(d; k). Our proof of Theorem I.1 in fact shows that given distinct shift spaces S(d; k) and S(d;k) of equal capacity, there exists a sliding-block map from S(d;k) onto S(d; k) if and only if one of Conditions 1-4 in the statement of the theorem holds. It follows that the only case where there exists a sliding-block map from S(d;k) onto S(d; k), and from S(d; k) onto S(d;k) as well, is when f(d;k); (d;k)g = f(0;1);(1; 1)g. Therefore, aside from S(0; 1) and S(1; 1), no pair of distinct (d; k)-constrained shift spaces can be conjugate. It is a well-known and trivial fact that S(0; 1) and S(1; 1) are indeed conjugate as shift spaces, the required conjugacy being obtained by mapping 0's and 1's to their respective complements. Thus, the only pair of (d; k)-constrained shift spaces that are conjugate are S(0; 1) and S(1; 1). Abstract-We give elementary recursive constructions of binary selforthogonal codes with dual distance four for all even lengths 12 and = 8. Consequently, good quantum codes of minimum distance three and four for such length are obtained via Steane's construction and the CSS construction. Previously, such quantum codes were explicitly constructed only for a sparse set of lengths. Almost all of our quantum codes of minimum distance three are optimal or near optimal, and some of our minimum-distance four quantum codes are better than or comparable with those known before. Index Terms-Binary code, quantum error correcting code, self-orthogonal code.