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Binary Construction of Quantum Codes of Minimum Distance Three and Four

R. Li, X. Li

2004
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IEEE Transactions on Information Theory
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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,
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... 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.

doi:10.1109/tit.2004.828149
fatcat:7iav3732hvfrfosm5udgq7jht4