Tight Bounds on the Minimum Euclidean Distance for Block Coded Phase Shift Keying
نویسنده
چکیده
We present upper and lower bounds on the minimum Euclidean distance dEmin(C) for block coded PSK. The upper bound is an analytic expression depending on the alphabet size q, the block length n and the number of codewords jCj of the code C. This bound is valid for all block codes with q 4 and with medium or high rate codes where jCj > (q=3). The lower bound is valid for Gray coded binary codes only. This bound is a function of q and of the minimum Hamming distance dHmin(B) of the corresponding binary code B. We apply the results on two main classes of block codes for PSK; Gray coded binary codes and multilevel codes. There are several known codes in both classes which satisfy the upper bound on dEmin(C) with equality. These codes are therefore best possible, given q; n and jCj. We can deduce that the upper bound for many parameters q; n and jCj is optimal or near optimal. In the case of Gray coded binary codes, both bounds can be applied. It follows for many binary codes that the upper and the lower bounds on dEmin(C) coincide. Hence, for these codes dEmin(C) is maximal.
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Generalized upper bounds on the minimum distance of PSK block codes
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