Encoding method and encoder for (n,n(n-1),n-1) permutation group code in communication modulation system

An encoding method for a (n, n(n−1), n−1) permutation group code in a communication modulation system, wherein the encoding method maps a k-length binary information sequence to a n-length permutation codeword in a signal constellation formed by the (n, n(n−1), n−1) permutation group code based on coset partition, and comprises following steps of:

• 1) constructing the (n, n(n−1), n−1) permutation group code, wherein when n is a prime number, the (n, n(n−1), n−1) permutation group code contains n(n−1) permutation codewords, each of the n(n−1) permutation codewords contains n code elements, a minimum Hamming distance between any two of the n(n−1) permutation codewords is n−1, and a code set Pn,xi of the (n, n(n−1), n−1) permutation group code is obtained by following expressions:
• Pn,xi=CnLn,xi={ci∘lj|ci∈Cn,lj∈Ln,xi,i∈Zn,j∈Zn−1} (1)
• ={(trn)n−1Ln,xi}={(tl1)n−1Ln,xi} (2)
• wherein the code set Pn,xi is obtained by an operator “∘” composition operation of a special cyclic subgroup Cn with a cardinality of |Cn|=n and a largest single fixed point subgroup Ln,xi with a cardinality of |Ln,xi|=n−1, ci represents an element in Cn, lj represents an element in Ln,xi, Zn represents a positive integer finite domain expressed by Zn={1,2, . . . , n}, Zn−1 represents a positive integer finite domain expressed by Zn−1={1,2, . . . , n−1}, and the code set Pn,xi has a cardinality of |Pn,xi|=n(n−1);
• the largest single fixed point subgroup Ln,xi is obtained by an expression (3):
• Ln,xi={a(l1,xi−xi)+xi|a∈Zn−1,xi∈Zn,l1,xi=[1 . . . n]} (3)
• wherein when n is a prime number, the largest single fixed point subgroup has a cardinality of |Ln,xi|=n−1;
• xi∈Zn indicates that the i-th code element of each of all n−1 permutation codewords in the largest single fixed point subgroup Ln,xi is a fixed point, and the other code elements are non-fixed points;
• and for xi=i and xi, i∈Zn, there are n fixed points in all, respectively corresponding to n largest single fixed point subgroups Ln,1, Ln,2, . . . , Ln,n=Ln;
• Pn,xi=CnLn,xi indicates that the code set Pn,xi is composed of n−1 cosets Cnl1,xi, Cnl2,xi, . . . , Cnln−1,xi of the special cyclic subgroup Cn;
• in the expression (2), the special cyclic subgroup Cn is replaced by a cyclic-right-shift operator function (trn)n−1 or a cyclic-left-shift operator function (tl1)n−1 to act on the largest single fixed point subgroup Ln,xi, so that the n−1 cosets of the special cyclic subgroup Cn are equivalently expressed as n−1 cyclic-left-shift orbits {(tl1)n−1l1,xi}, {(tl1)n−1l2,xi}, . . . , {(tl1)n−1ln−1,xi} or cyclic-right-shift orbits {(trn)n−1l1,xi}, {(trn)n−1l2,xi}, . . . , {(trn)n−1ln−1,xi};
• for an equivalent operation of {(trn)n−1Ln,xi} and {(tl1)n−1Ln,xi}, a coset leader or orbit leader array composed of n−1 permutation codewords is first calculated by the expression (3) for calculating the largest single fixed point subgroup Ln,xi, and then all n(n−1) codewords of the code set Pn,xi are calculated by the expression (2);
• for each orbit, a permutation codeword lj,xi of the largest single fixed point subgroup Ln,xi is input to a n-dimension cyclic shift register to perform n−1 cyclic-left-shift operations, equivalent to performing {(trn)n−1lj,xi}, or to perform n−1 cyclic-right-shift operations, equivalent to performing {(trn)n−1lj,xi}, thereby generating n permutation codewords;
• the n(n−1) permutation codewords are generated by n−1 orbits;
• 2) selecting 2k permutation codewords from the n(n−1) permutation codewords of the code set Pn,xi to form a signal constellation Γn such that the signal constellation Γn still has coset characteristics, that is, the signal constellation Γn contains the 2k permutation codewords, and is partitioned into 2k1 cosets, and each coset of the 2k1 cosets contains 2k2 permutation codewords, wherein k=k1+k2, 2k≤n(n−1), 2k1≤n−1, 2k2≤n, and an exact value of k is k=└log2 n(n−1)┘ and satisfies |Γn|=2k;
• 3) defining a mapping function φ:
• Hk→Γn by a function π=φ(h), and mapping, by the mapping function φ, a k-length binary information sequence h=[h1h2 . . . hk] in a set Hk of 2k k-length binary information sequences to a signal point π=[a1a2 . . . an] in the signal constellation Γn composed of the |Γn|=2k n-length permutation codewords, wherein π∈Γn, h∈Hk, h1, h2, . . . , hk∈Z2={0,1}, and a1, a2, . . . , an∈Zn={1,2, . . . , n}.