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

Document type: Researchreports
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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
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