The Construction of Permutation Group Codes for Communication Systems: Prime or Prime Power?

This paper focuses on correcting a theorem relating to the construction of permutation group codes (PGCs). The theorem in question assumed that affine transformation could be used to enumerate all the code words of a permutation array with a minimum Hamming distance (MHD) of
$n-1$
for any
$n >1$
. This assumption was founded upon the proposition that, if the code length,
$n$
, is a prime power, then the maximum cardinality of the code will be
$n(n-1)$
and its MHD will be
$n-1$
. However, two typical algebraic methods, one relying on affine transformation and another upon the composite operation of two small subgroups of a symmetric group, can violate this proposition. This is because it is only when
${n}$
is a prime rather than a prime power that they can enumerate all the code words of an
$(n;n\left ({n-1}\right);n-1)$
-PGC. By investigating how the range of
$n$
impacts upon the cardinality and MHD of a code, we provide a corrective theorem that stipulates the construction of
$(p^{q};p^{2q}(1-1/p);p^{q}(1-1/p))$
-PGCs when
$n=p^{q}$
is a prime power, where
$p$
is a prime and
$q>1$
. On the basis of this theorem, and under the condition of
$n$
being the power of 2, we construct a
$(2^{q};2^{2q-1};2^{q-1})$
-PGC and present an encoder that can map a
$k$
-bit binary information sequence to an
$n=2^{2q}$
-dimension permutation code word. The natural array structure of
$(2^{q};2^{2q-1};2^{q-1})$
-PGCs makes them especially well-suited to forming the basis of low-complexity encoders. We present simulation-based experiments that show that, as the code length increases, the performance of these codes improves. The best performance is for a (16;128;8)-PGC, which can achieve −3.8 dB with a word error rate of 10
⁻⁷
.