Hermitian forms, trace equations and application to codes Dany-Jack Mercier∗ November 14th, 2001 Abstract: We provide a systematic study of sesquilinear hermitian forms and a new proof of the calculus of some exponential sums defined with quadratic hermitian forms. The computation of the number of solutions of equations such as TrFt/Fs (f (x)+v.x) = 0 or TrFt/Fs (f (x)) = a allows us to construct codes and to obtain their parameters. Key-Words: Sesquilinear Hermitian Forms, Quadratic Forms, Finite Fields, Traces, Exponential Sums, Codes. MSC-Class: 94B05, 11T23, 11T71. Versions: Minor changes in version 2 (21th april 2003). 1 Introduction Let Ft be the finite field with t elements and characteristic p. Our first purpose is to adapt the arXiv:math/0111191v2 [math.NT] 21 Apr 2003 C F 2 classical hermitian form Theory on to the case of the finite field t , considering the involution t F 2 x 7→ x in t instead of the application z 7→ z. Then we introduce exponential sums associated with quadratic hermitian forms and obtain the number of solutions of certain equations on Ft. At this point it is easy to construct two linear codes using the same method as Reed-Muller codes, to get their parameters and to compare them to the classical Reed-Muller construction. A general introduction to hermitian forms over a finite field is given by Bose and Chakravarti in [1], and the use of those objects in coding Theory has been discussed for instance in [3], [4] or [5]. ∗NT/0111191 [ccof0007] v 1.00 Equipe Applications de l’Alg`ebre et de l’Arithm´etique, D´epartement de Math´ematiques & Informatique, Universit´edes Antilles-Guyane, Campus Fouillole, F97159 Pointe-`a-Pitre cedex, France, e-mail : [email protected]. 1 My first contribution will be to review in details the main results of [1] and [3], giving alternative proofs of some important results in the Theory. It is worth pointing out that the existence of a H-orthogonal basis is shown by induction without explicit calculus on rows and columns of a matrix as in [1] (see Theorem 4). Another example is given by a new and staightforward proof of the most precious result of the paper of Cherdieu [3] (see Theorem 14). Section 10 is devoted to the construction of the code Γ of [3], and Section 11 use the same argument to give another example of such a linear code. As this paper wants to remain self- contained, an annex in Section 12 will summarize without proofs the relevant material on characters on a finite group. FN 2 Sesquilinear hermitian forms on t2 FN Let N be an integer ≥ 1 and let E be the vector space t2 . FN FN F 2 FN D´efinition 1 A function H : t2 × t2 → t is a sesquilinear form on E = t2 if it is semi-linear in the first variable and linear in the second variable, i.e. F 2 ′ ′ t t ′ (1) ∀λ, µ ∈ t ∀x,x ,y ∈ E H (λx + µx ,y)= λ H (x,y)+ µ H (x ,y) , F 2 ′ ′ ′ (2) ∀λ, µ ∈ t ∀x,y,y ∈ E H (x, λy + µy )= λH (x,y)+ µH (x,y ) . The sesquilinear form H is called hermitian if (3) ∀x,y ∈ E H (x,y)= H (y,x)t . A sesquilinear hermitian form will be called a hermitian form on E. The vector space of all FN FN hermitian forms on t2 will be denoted by H t2 . Note that properties (2) and (3) give (1), and that H (x,x) ∈ Ft for all x ∈ E as soon as H is a hermitian form. F 2 t F 2 If x ∈ t we put x = x and we say that x is the conjugate of x. If α ∈ t satisfies F 2 F F 2 F t = t (α), each element x of t is uniquely written as x = a + bα with a and b in t. Then x = (a + bα)t = a + bαt = a + bα. F 2 D´efinition 2 Let A = (aij)i,j be a square matrix with i, j = 1, ..., N and with entries in t . T We denote by A the matrix A = (aij)i,j and by A the transpose of A. The conjugate of ∗ T t ∗ A = (aij) is the matrix A = A = aji . The matrix A is hermitian if A = A. i,j i,j If e = (e1, ..., eN ) is a basis of E and if H is a sesquilinear form on E, H (x,y)= H x e , y e = x y H (e , e )= X∗MY i i j j i j i j Xi Xj Xi,j T T where M = (H (ei, ej))i,j, X = (x1, ..., xN ) and Y = (y1, ..., yN ). We say that M is the matrix of H in the basis e, and we write M = Mat (H; e). Th´eor`eme 1 A sesquilinear form H is hermitian if, and only if, its matrix Mat (H; e) in a basis e is hermitian. 2 Proof : Let M denotes the matrix Mat (H; e). If H is a hermitian form, H (ei, ej)= H (ej, ei) implies M ∗ = M. Conversely, M ∗ = M implies ∗ ∀X,Y H (Y, X)= Y M X = T Y M X = T T Y M X = X∗MY = H (X,Y ) . N 2 F F 2 Corollaire 1 dim t H t = N . N 2−N Proof : The matrix H = (aij) of a hermitian form depends on coefficients aij (where 2 2 F 2 F N −N 2 1 ≤ j < i ≤ N) in t and N coefficients aii (1 ≤ i ≤ N) in t. Thus we have 2× 2 +N = N independent parameters. e′ Let Pe = P denotes a change of coordinates from a basis e = (e1, ..., eN ) to another basis ′ ′ ′ T ′ T ′ ′ e = (e1, ..., eN ). Let X = (x1, ..., xN ) and X = (x1, ..., xN ) stands for the N-tuplets of coordinates of the same vector in basis e and e′. Then ∗ H (X,Y )= X∗MY = P X′ M PY ′ = X′∗ (P ∗MP ) Y ′ and Mat (H; e′)= P ∗MP is the matrix of H in the new basis e′. 3 Kernel and rank of H If x ∈ E and if H denotes a hermitian form, we define the linear application H (x, .) in the dual E∗ by: F 2 H (x, .) : E −→ t y 7−→ H (x,y) . The map: H : E −→ E∗ x 7−→ H (x, .) e ′ ′ ′ F 2 is semi-linear, i.e. satisfies H (λx + x ) = λH (x)+ H (x ) for all vectors x, x and all λ ∈ t . The extern law • defined by λ • l = λ.l gives us a new vector space structure on E∗. For convenience, we shall writeeE∗ instead of eE∗ whene we use this new extern law. The map H : E → E∗ is semi-linear if, and only if, H : E → E∗ is linear, and we can introduce the ∗ ∗ ∗ ∗ matrix of H in the basis e = (e1, ..., eN ) in E and the dual basis e = (e1, ..., eN ) in E . The linearitye of H gives us the same results as ine the case of symmetric bilinear forms. Namely: e Th´eor`emee 2 The equality Mat H; e, e∗ = Mat (H; e) holds for all hermitian form H. e∗ ∗ Proof : Assume that Mat H; e, e = (aij). Then H (ej)= aij • ei and i e e P H (ej) (ek)= H (ej, ek)= akj. Thus Mat H; e, e∗ = T Mat (He; e) = Mat (H; e). D´efinitione 3 The kernel Ker H (resp. rank rk H) of H is the kernel (resp. rank) of H. Thus Ker H = {x ∈ E / ∀y ∈ E H (x,y) = 0} and rk H = rk (Mat (H; e)). e 3 4 Orthogonality D´efinition 4 Let H denotes a hermitian form on E. Vectors x and y are orthogonal if H (x,y) = 0. If F is a subset of E, the subspace F ⊥ = {x ∈ E / ∀y ∈ F H (x,y) = 0} is called the orthogonal of F . It is easily seen that: Th´eor`eme 3 For all subspaces F and G in E, ⊥ F ⊂ F ⊥ , F ⊂ G ⇒ G⊥ ⊂ F ⊥, (F + G)⊥ = F ⊥ ∩ G⊥, (F ∩ G)⊥ ⊃ F ⊥ + G⊥. D´efinition 5 A basis e = (e1, ..., eN ) is orthogonal for the hermitian form H (we say H- orthogonal) if H (ei, ej ) = 0 when i 6= j. This means that the matrix Mat (H; e) is diagonal. Lemme 1 Let H denotes a sesquilinear form on E. If t is odd and if q (x) = H (x,x), then for all x,y in E, 1 1) H (x,y)+ H (y,x)= 2 [q (x + y) − q (x − y)] , 1 2) H (x,y) − H (y,x)= α−α [q (x + αy) − q (x + αy)] , 1 1 3) H (x,y)= 4 [q (x + y) − q (x − y)] + 2(α−α) [q (x + αy) − q (x + αy)] . Proof : We have q (x + y) − q (x − y)= q (x)+ H (x,y)+ H (y,x)+ q (y) − [q (x)+ H (x, −y)+ H (−y,x)+ H (−y, −y)] = 2 (H (x,y)+ H (y,x)) and q (x + αy) − q (x + αy)= q (x)+ αH (x,y)+ αtH (y,x)+ αt+1q (y) − q (x)+ αtH (x,y)+ αH (y,x)+ αt+1q (y) t = α − α (H (x,y) − H (y,x)) . t As α∈ / Ft, we have α 6= α and the Lemma follows Lemme 2 The norm application ∗ ∗ F m F NFtm /Ft : t → t m−1 x 7→ xt +...+t+1 m is a multiplicative group epimorphism, and N −1 (b) = t −1 for all b ∈ F∗.