By Cunsheng Ding

This can be the 1st monograph on codebooks and linear codes from distinction units and virtually distinction units. It goals at delivering a survey of structures of distinction units and virtually distinction units in addition to an in-depth therapy of codebooks and linear codes from distinction units and virtually distinction units. To be self-contained, this monograph covers beneficial mathematical foundations and the fundamentals of coding concept. It additionally comprises tables of most sensible BCH codes and top cyclic codes over GF(2) and GF(3) as much as size one hundred twenty five and seventy nine, respectively. This repository of tables can be utilized to benchmark newly built cyclic codes. This monograph is meant to be a reference for postgraduates and researchers who paintings on combinatorics, or coding conception, or electronic communications.

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Commutative presemifields can be utilized to construct planar functions, as demonstrated by the next theorem [Kantor (2003)]. 25. Let (K , +, ×) be a finite presemifield with commutative multiplication. Then the function f (x) = x × x is a planar function from (K , +) to itself. To introduce planar monomials on GF(q 2 ) from commutative semifields in the sequel, we do the following preparations. Let {1, β} be a basis of GF(q 2 ) over GF(q). Let x = x 1 + x 2 β, where x i ∈ GF(q). It is easily seen that x1 = β q x − βx q xq − x , x2 = q , q β −β β −β where q is a prime power.

For periodic sequences, there are a few ways to determine their linear span and minimal polynomials. 2]. 16. Let λ∞ be a sequence of period n over GF(q). The generating i polynomial of λ∞ is defined by n (x) = n−1 i=0 λi x ∈ GF(q)[x]. 12) n (x))). The other one is given in the following lemma [Antweiler and Bomer (1992)]. 17. Any sequence λ∞ over GF(q) of period q m − 1 has a unique expansion of the form q m −2 λt = ci α it for all t ≥ 0, i=0 where ci ∈ GF(q m ). Let the index set be I = {i : ci = 0}, then the minimal polynomial Mλ (x) of λ∞ is (1 − α i x), Mλ (x) = i∈I and the linear span of λ∞ is |I |.

79]. The Gaussian periods are also determined in the semiprimitive case and are described in the next theorem. 18 ([Baumert, Mills and Ward (1982)]). Assume that p is a prime, N ≥ 2 is a positive integer, r = p 2 j γ , where N|( p j + 1) and j is the smallest such positive integer. Then the Gaussian periods of order N are given below: (a) If γ , p, p j +1 N are all odd, then √ √ √ 1+ r N r +1 (N,r) η(N,r) = r − = − , η for all i = . N/2 i N N 2 (b) In all the other cases, √ √ (−1)γ r − 1 , η0(N,r) = −(−1)γ r + N √ (−1)γ r − 1 (N,r) for all i = 0.