On the Hyper-Kloosterman Codes Over Galois Rings

The Hyper-Kloosterman code was ﬁrst deﬁned over ﬁnite ﬁelds by Chinen-Hiramatsu, see (Chinen, & Hiramatsu, 2001)


Introduction
It all starts with a paper by Kloosterman in 1926(Kloosterman, 1926, in his study of certain positive definite integral quadratic forms. In his paper, Kloosterman, introduced certain exponential sum (since then known as the Kloosterman sum). Since then Kloosterman sums have enjoyed much attention of the finite fields. Some of this interest is due to their applications in cryptography and coding theory; see for example: the search for the number of solution of certain equations over finite fields, the distribution of values of Kloosterman sums, the divisibility properties of classical binary Kloosterman sums , see (Pascale, Helleseth , & Victor, 2009) , the search for the weight of certain codes, called the Kloosterman code, see (Jacques,1989), (Gilles, 1989) and (Chinen, & Hiramatsu, 2001). The hyper-Kloosterman code was first defined over finite fields by Chinen-Hiramatsu, (Chinen, & Hiramatsu, 2001). In the present paper we define the Hyper-Kloosterman codes over Galois rings R(p e , m). We show that this code is the trace of linear code over R(p e , m). By the Hyper-Kloostermann sums over Galois rings, we determine the Hamming weight of any codeword of this code over Galois rings.

Preliminaries
Some preliminaries on Galois rings are given below. For more details, the reader is referred to (Shuqin, & Han, 2004). Let e ≥ 1 be a fixed integer, p a prime number. A monic polynomial h(x) ∈ Z p e [x] is said to be a basic irreducible polynomial of degree m if (h(x) mod p)∈ Z p [x] is a monic irreducible polynomial of degree m. Galois ring R e,m = GR(p e , m) is the unique unramified extension of degree m over Z p e and can be written as where h(x) is a basic irreducible polynomial of degree m over Z p e . The ring R e,m is a local ring with unique maximal ideal pR e,m . The unit set R * e,m = R e,m \pR e,m in R e,m is a multiplicative group of order ♯R * e,m = (p m − 1)p m(e−1) = p me − p m(e−1) . The set R * e,m always contains a cyclic group of order p m −1. In analogy with finite fields, we will call an element a primitive element of the Galois ring R e,m if it is a generator for this cyclic group. Let γ e,m denote a primitive element in R e,m . Let T * = {1, γ e,m , γ 2 e,m , ..., γ p m −2 e,m }. The set T * is called Teichmuller system. Let T = T * ∪ {0} = {0, 1, γ e,m , γ 2 e,m , ..., γ p m −2 e,m } . For any element z ∈ R e,m the p-adic expansion has giving by: Let τ the Frobenius map of R e,m over Z p e given by where z = ∑ e−1 i=0 p i z i ∈ R e,m and z i ∈ T . As we know τ is the generator of Galois group of R e,m /Z p e which is a cyclic group of order m. The trace mapping tr e,m : R e,m → Z p e is defined via

Hyper-Kloosterman Codes
Definition 3.1 In a similar way to the Hyper-Kloosterman codes over Galois fields defined for the first time by Chinen-Hiramatsu , we define the hyper-Kloosterman code C l (p e , m) of degree l − 1, (l ≥ 2, l is an integer ), over Galois ring, by the image of the map: e,m ) l−1 represents a vector obtained by letting x run through the set (R * e,m ) l−1 (such a notation is often used in the literature on the trace codes). The code C l (p e , m) is a generalization, over Galois ring, of the hyper-Kloosterman code. The Hyper-Kloosterman code over Galois fields has been investigated by many authors. See for example: (Chinen, & Hiramatsu, 2001), (Chinen, 2003) and (Moisio, 2008).
Remark 3.1 The length of a code C l (p e , m) is a power of the order of the group of invertible elements, R * e,m .
Proof φ l is a module homomorphism and so his image is a sub-module of Z (p me −p m(e−1) ) l−1 p e . Therefore C l (p e , m) is a linear code. Proof Let α l the map defined by: a = (a 1 , ..., a l ) ∈ R l e,m and x = (x 1 , ..., x l−1 ) ∈ (R * e,m ) l−1 . We see easily α l is a module homomorphism , and so his image is a sub-module. Let C l (p e , m) = Im(α l ), the image of α l . C l (p e , m) is a linear code over Galois ring R e,m . By definition of α l , we obtained C l (p e , m) = tr e,m (C l (p e , m)).
In the following, C l (p e , m) denote the code defined in the previous proof. That is to say:  (p e , m)).
Let f 1 , f 2 , ..., f (p me −p m(e−1) ) l−1 be a fixed ordering of the elements of. Moreover we define I(x) for x = (x 1 , x 2 ..., x l−1 ) ∈ (R * e,m ) l−1 by: Let f T j the transpose of the vector f j . Then we form the matrix as follows: ) is a generator matrix of the code C l (p e , m)

Proof
Let a = (a 1 , ..., a l ) ∈ R l e,m aG l (p e , m) = (a 1 , ..., a l ) Which is a codeword of the code C l (p e , m).
Therefore, the matrix G l (p e , m) is a generator matrix of the code C l (p e , m).

Hamming Weights of the Codewords of C l (p e , m)
Definition 4.1 Let C a code of length n. x = x 1 x 2 . . . x n and y = y 1 y 2 . . . y n two codewords on code C. The Hamming distance of x and y, is given by The Hamming weight W H of a codeword x = x 1 x 2 . . . x n of C is the number of non-zero x i for 1 ≤ i ≤ n.
Definition 4.2 Additive characters over Galois rings.
An additive character of R e;m is a homomorphism from the additive group of R e;m to C * ; the multiplicative group of complex field. We Define ψ(a) = exp(2πitr e,m (a)/p e ) for any given element a in R e;m and " exp" denote the exponential function : It is easily seen that ψ is an additive character of R e;m ; called the canonical additive character. For b ∈ R e;m ; define ψ b (a) = ψ(ba): a ∈ R e;m . ψ is also an additive character. In fact, we have: We present a well-known result, sometimes called the orthogonality of characters, as a proposition for later reference.  Vol. 12, No. 2;2020 where T r(a, x) = tr e,m (a 1 x 1 + a 2 x 2 + ... + a l−1 x l−1 + a l (x 1 x 2 ...x l−1 ) −1 ) for a = (a 1 , a 2 , ..., a l ) and x = (x 1 , x 2 , ..., x l−1 ) , and tr e,m = traceR e,m /Z p e The Kloosterman sums have been used by several authors to evaluate the Hamming weights of a certain linear code C(q), called the Kloosterman code (see for example (Gilles, 1989) and (Jacques, 1989) ).
Theorem 4.1 For any codeword φ l (a) ∈ C l (p e , m), the weight of φ l (a) is given by: Proof.

Conclusion and Perspectives
We have generalized the hyper-Klosterman codes over Galois rings. This code is seen as a trace of a linear code over galois rings. We get the Hamming weight of a codeword by the hyper-kloosterman sums. The first trivial example gives the 1-quasi-cyclic or cyclic codes. We can therefore continue to study the properties of this code, quasy-cyclic properties and others.