Tight Bounds on the Minimum Euclidean Distance for Block Coded Phase Shift Keying
| Document type: | Researchreports |
|---|---|
| Full text: | |
| Author(s): | Magnus Nilsson, Håkan Lennerstad |
| Title: | Tight Bounds on the Minimum Euclidean Distance for Block Coded Phase Shift Keying |
| Series: | Research Report |
| Year: | 1996 |
| Issue: | 15 |
| ISSN: | 1103-1581 |
| Organization: | Blekinge Institute of Technology |
| Department: | Dept. of Telecommunications and Mathematics (Institutionen för telekommunikation och matematik) Dept. of Telecommunications and Mathematics S-371 79 Karlskrona +46 455 780 00 http://www.hk-r.se/itm/index.html |
| Authors e-mail: | magnus.nilsson@te.hik.se, hakan@itm.hk-r.se |
| Language: | English |
| Abstract: | 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 bound is 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 many parameters $q,n$ 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. |
| Subject: | Mathematics\Discrete Mathematics Telecommunications\Coding Theory |
| Keywords: | Block codes, phase shift keying, minimum Euclidean distance, multilevel codes, coded modulation, Gray code, non-linear codes. |
| Note: | This is a revised version of the printed Research Report 15/96. Magnus Nilsson is assistant professor in telecommunications, Univ. of Kalmar, Sweden. |
| URN: | urn:nbn:se:bth-00032 |
