User-distinguished finite-field resource construction method and finite-field multiple access system

A user-distinguished finite-field resource construction method, executed by a processor, wherein the construction method is a basic-field resource construction method and comprises the following steps:

• let GF(p) be a prime field, and p−1 nonzero elements in the field are divided into (p−1)/2 mutually-disjoint element pairs EP, wherein each EP comprises two elements, k and additive inverse p−k in the field, and the element pair (k, p−k) is referred to as additive inverse element pair AIEP;
• such division is denoted as P;
• let J be a positive integer not greater than (p−1)/2, namely, 1≤J≤(p−1)/2;
• let C1,C2, . . . , CJ be J AIEPs in the P, and thus Cj=(tj,p−tj);
• wherein 1≤j≤J, 1≤tj≤p;
• tj is a nonzero element in GF(p), and C refers to a set {C1,C2, . . . , CJ};
• let (u1,u2, . . . , uJ) be one J tuple on GF(p), wherein uj is one element in Cj;
• and thus the J tuple (u1,u2, . . . , uJ) is one element in Cartesian product C1×C2× . . . ×CJ;
• the mod p of each element and τ=⊕j=1Juj in the tuple are referred to as finite-field sum-pattern (FFSP) of the J tuple;
• wherein ⊕ represents modular addition operation;
• the J tuple (p−u1, p−u2, . . . , p−uJ) is also one element in Cartesian product and its sum pattern is p−τ=p−⊕j=1Juj;
• if the sum pattern of the (u1,u2, . . . , uJ) is 0, the sum pattern of the (p−u1, p−u2, . . . , p−uJ) is also 0;
• let (u1,u2, . . . , uJ) and (u′1,u′2, . . . , u′J) be two J tuples in C1×C2× . . . ×CJ;
• if ⊕j=1Juj≠⊕j=1Ju′j, one sum pattern uniquely specifies one J tuple in C1×C2× . . . ×Cj, namely,
• (
• u
• 1
• ,
• u
• 2
• ,
• …
• ,
• u
• J
• )
• ↔
• ⊕
• j
• =
• 1
• J
• u
• j
• is one one-to-one mapping;
• given one sum pattern τ=⊕j=1Juj, a corresponding J tuple (u1,u2, . . . , uJ) is uniquely restored;
• the C1×C2× . . . ×CJ is said to have a unique sum pattern mapping structure USPM, and J≤log2(p−1);
• for the set C={C1,C2, . . . , CJ} formed by J AIEPs, if the set has a USPM structure, the set serves as a finite-field resource in multiple access communication;
• in one multiple access communication system, J AIEPs are assigned to J users;
• if each user respectively transmits a corresponding AIEP symbol, a transmitter collects J transmission symbols, obtains finite-field sum-pattern (FFSP) τ=⊕j=1Juj and sends the sum pattern over a multiple access channel;
• the J transmission symbols are restored uniquely and unambiguously by the sum pattern τ;
• if each J tuple is seen as codewords of J users, the set C=C1×C2× . . . ×CJ forms one J user code totaling 2J codewords in GF(p);
• C=C1×C2× . . . ×CJ is said to be one AIEP code uniquely decodable by the J users in GF(p), which is denoted as J user UDAIEP code;
• when this code is used in a J user multiple access communication system, the symbol sent by the j-th user corresponds to the j-th component ui of the codewords.