Tight Bounds on the Minimum Euclidean Distance for Block Coded Phase Shift Keying

Document type: Researchreports Yes STG44714STG38246Research Report 1996_15.pdf Magnus Nilsson, Håkan Lennerstad Tight Bounds on the Minimum Euclidean Distance for Block Coded Phase Shift Keying Research Report 1996 15 1103-1581 Blekinge Institute of Technology Dept. of Telecommunications and Mathematics (Institutionen för telekommunikation och matematik)Dept. of Telecommunications and Mathematics S-371 79 Karlskrona+46 455 780 00http://www.hk-r.se/itm/index.html magnus.nilsson@te.hik.se, hakan@itm.hk-r.se English We present upper and lower bounds on the minimum Euclidean distance \$d_{Emin}(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 \$|C|\$ of the code \$C\$.This bound is valid for all block codes with \$q\geq4\$ and with medium or high rate - codes where \$|C|>(q/3)^n\$. The lower bound is valid for Gray coded binary codes only. This boundis a function of \$q\$ and of the minimum Hamming distance \$d_{Hmin}(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 \$d_{Emin}(C)\$ with equality. These codes are therefore best possible, given \$q,n\$ and \$|C|\$. We can deduce that the upper bound for manyparameters \$q,n\$ and \$|C|\$ 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 \$d_{Emin}(C)\$ coincide. Hence, for these codes \$d_{Emin}(C)\$ is maximal. Mathematics\Discrete MathematicsTelecommunications\Coding Theory Block codes, phase shift keying, minimum Euclidean distance, multilevel codes, coded modulation, Gray code, non-linear codes. This is a revised version of the printed Research Report 15/96. Magnus Nilsson is assistant professor in telecommunications, Univ. of Kalmar, Sweden. urn:nbn:se:bth-00032