Skew Constacyclic Codes over Galois Rings

Transcription

1 Skew Constacyclic Codes over Galois Rings Delphine Boucher Patrick Solé Felix Ulmer Delphine Boucher IRMAR (UMR 6625) Université de Rennes 1, Campus de Beaulieu, F Rennes Patrick Solé I3S 2000 route des Lucioles, F Sophia Antipolis Felix Ulmer IRMAR (UMR 6625) Université de Rennes 1, Campus de Beaulieu, F Rennes Abstract We generalize the construction of linear codes via skew polynomial rings by using Galois rings instead of finite fields as coefficients. The resulting non commutative rings are no longer left and right Euclidean. Codes that are principal ideals in quotient rings of skew polynomial rings by a two sided ideals are studied. As an application, skew constacyclic self-dual codes over GR(4, 2) are constructed. Euclidean self-dual codes give self-dual Z 4 codes. Hermitian self-dual codes yield 3 modular lattices and quasi-cyclic self-dual Z 4 codes. Introduction Polynomial rings and their ideals are essential to the construction and understanding of cyclic codes. For the first time in [5] non commutative skew polynomial rings have been used to construct (a generalization of) cyclic codes. In that approach, it is necessary to use as alphabet a finite field with a non trivial Galois automorphism, like, e.g. F 4. In the present work we extend that approach by considering as alphabets Galois rings of even characteristic like, e.g. GR(4, 2). The technical difficulty in passing from field alphabet to ring alphabet is that the skew polynomial rings may not be Ore rings; in particular they are no longer left and right Euclidean. However, left and right divisions by unitary polynomials are still well defined. Therefore codes that are principal ideals generated by unitary polynomials in quotient rings of skew polynomial rings by a two sided ideal are studied. In particular the problem of finding central polynomials, that is generators of principal two sided ideals is addressed. As an application self-dual codes over GR(4, 2) are constructed, and used for three of the four applications of [14]. 1)Self-dual Euclidean codes give self-dual Z 4 codes by projection on a trace orthogonal basis. Many Type I codes in length 24 are obtained and classified by the root systems of their Construction A lattice. New coding constructions of the Odd Leech lattice, the only unimodular lattice of norm 3 in dimension 24 are given thus supplementing the results of [15]. Some new Type II codes in length 24 are also obtained, that yield the Leech lattice by Construction A, though not 1

2 being Lee-optimal in the sense of Rains [20]. A Type I code in length 40 is constructed that is not Type IV. In fact its Lee and Euclidean distances are better than what would be possible for a Type IV code of that length [9]. 2)Self-dual Hermitian codes build 3-modular lattices. In particular a simpler construction of one of the two extremal such lattices in dimension 28 is given. 3)Self-dual Hermitian codes yield self-dual quasi-cyclic codes over Z 4 by the cubic construction. Self-dual Type II codes are obtained in length 24 and classified by their symmetrized weight enumerator and root system. The material is organized as follows. Section 1 contains generalities on skew polynomial rings over Galois rings. Section 2 defines the codes generated by principal ideals in quotient rings where X n 1 is replaced by a central polynomial. Section 3 explains how to generate central polynomials by using the notion of bound and gives a few examples. Section 4 considers parity check matrices and duals of the constacyclic codes defined before. Section 5 constructs Euclidean self-dual codes over GR(4, 2). Section 6 is devoted to self-dual Z 4 codes and their unimodular lattices. Section 7 studies Hermitian self-dual codes over GR(4, 2) and 3 modular lattices. Section 8 considers the cubic construction of self-dual Z 4 codes from Hermitian self-dual codes over GR(4, 2). 1 Skew polynomials over Galois rings and their quotients Let p N be a prime number and s N a positive integer. A Galois ring GR(p s, m) is a Galois extension of Z/(p s ) of degree m N. The properties of GR(p s, m) that we summarize in what follows, can be found in [19]. The ring GR(p s, m) is isomorphic to Z/(p s )[Y ]/(h), where h Z/(p s )[Y ] is a monic polynomial of degree m whose reduction in Z/(p)[Y ] is irreducible. Galois rings with the same parameters p, s and m are isomorphic. The ring GR(p s, m) is a commutative local ring with maximal ideal (p) and residue field F p m. We denote ξ the image of Y in Z/(p s )[Y ]/(h) and obtain GR(p s, m) = Z/(p s )[ξ]. Any element of GR(p s, m) has a unique expression of the form α 0 + α 1 ξ α m 1 ξ m 1 (α i Z/(p s )). The image ξ of ξ in GR(p s, m)/(p) = F p m is a root of h whose multiplicative order is p m 1. The set T = {0, 1, ξ,..., ξ pm 2 } is called the Teichmüller set of GR(p s, m). Any element of GR(p s, m) also has a unique expression of the form β 0 + p β p s 1 β s 1 (β i T ). The above representation is called the p-adic expansion of the element. We denote by θ the Frobenius map θ: GR(p s, m) GR(p s, m) s 1 p i β i s 1 p i (β i ) p (β i T ). The group of automorphisms Aut(GR(p s, m)) of the Galois ring GR(p s, m) is cyclic of order m and is generated by θ. The Frobenius automorphism θ Aut(GR(p s, m)) fixes pointwise the subring GR(p s, 1) = Z/(p s ) of GR(p s, m). In the following we denote by GR(p s, m) the set of invertible elements and we use the fact that this set is left invariant by all automorphisms θ i Aut(GR(p s, m)). 2

3 One defines a ring structure on the set GR(p s, m)[x, θ] = {a n X n a 1 X + a 0 a i GR(p s, m) and n N} of formal polynomials where the coefficients are written on the left of the variable X. The addition in GR(p s, m)[x, θ] is defined to be the usual addition of polynomials and the multiplication is defined by the basic rule Xa = θ(a)x (a GR(p s, m)) and extended to all elements of GR(p s, m)[x, θ] by associativity and distributivity. Note that two polynomials are equal if and only if all their coefficients are equal. Lemma 1.1 The center Z(GR(p s, m)[x, θ]) of GR(p s, m)[x, θ] is Z/(p s )[X m ]. Proof. The subring of the elements of GR(p s, m) that are fixed by θ is Z/(p s ). For any integer i N, the power X i m is also in the center Z(GR(p s, m)[x, θ]) of GR(p s, m)[x, θ]. This follows from the fact that m is the order of the automorphism θ, showing that for any a GR(p s, m) we have X i m a = (θ m ) i (a)x i m = ax i m. This shows that f = α 0 + α 1 X m + α 2 X 2m α s X s m with α i Z/(p s ) is a central element. Conversely, for f in Z(GR(p s, m)[x, θ]), considering Xf fx and af fa for a GR(p s, m) one proves that f Z/(p s )[X m ]. Since GR(p s, m) contains zero divisors, many properties of skew polynomial rings over fields are no longer true : Example. The following are two distinct factorizations of X 4 1 in GR(4, 2)[X, θ] into irreducible monic polynomials: (X + 1)(X + 1)(X + 2ξ + 1)(X + 2ξ + 3) and (X 2 + 2ξ + 1)(X 2 + 2ξ + 3). This shows that the degrees in different factorizations of the same polynomial are not unique up to permutation. The ring GR(p s, m)[x, θ] is no longer left or right Euclidean, but left or right division can be defined for some elements. Consider f = s a ix i and g = t j=0 b jx j. As usual we set the degree of 0 to be. If s t and the leading coefficient b t of g is invertible, then: 1. We can define a right division of f by g. We simply note that the degree of f a s θ s t (b t ) Xs t g is less than the degree of f. To prove this, it is sufficient to compute the leading coefficients of both polynomials and see that they cancel. In the above we use the fact that θ s t is also an automorphism and that the image of an invertible element is invertible. Iterating the above by subtracting further left multiples of g from the result until the degree is less than the degree of g, we obtain polynomials q and r such that : f = q g + r and deg(r) < deg(g) If r = 0 we say that g is a right divisor of f. 2. Similarly to the right division, we can define a left division of f by g using the fact that the degree of ( ( ) ) f g θ t as X s t b t is less than the degree of f. To prove this, it is again sufficient to compute the leading coefficients of both polynomials and see that they cancel. In the above we use the fact that θ t is also an automorphism and that the image of b t is again an invertible element. 3

4 Iterating the above by subtracting further right multiples of g from the result until the degree is less than the degree of g, we obtain polynomials q and r such that : If r = 0 we say that g is a left divisor of f. f = g q + r and deg(r) < deg(g) We will also need the fact that the remainder of a division of f by a monic polynomial g GR(p s, m)[x, θ] is unique. Suppose are two right divisions by g, then f = q 1 g + r 1 = q 2 g + r 2 (q 1 q 2 ) g = r 2 r 1. If q 1 q 2 is not zero, then the right polynomial is of degree at least the degree of g, while the right polynomial is of degree at most one less than the degree of g. Therefore q 1 = q 2, from which we get r 2 = r 1. The proof for left division is similar. Example. obtained via In GR(4, 2)[X, θ] the polynomial X ξ is a right divisor of X 2 1. This is (X 2 1) X(X ξ) = X 2 1 X 2 + Xξ = ξ 2 X 1 (here Xξ = θ(ξ)x = ξ 2 X) and in the next step (ξ 2 X 1) ξ 2 (X ξ) = ξ 2 X 1 ξ 2 X + ξ 3 = 0 Therefore the remainder is 0 and the left quotient (of the right division) is X + ξ 2. We get X 2 1 = (X + ξ 2 ) (X ξ) Note that not all left or right ideals in GR(p s, m)[x, θ] are principal, but in the following we will focus on those ideals. 2 Codes defined by principal ideals with monic generator Proposition 2.1 A left or right ideal in GR(p s, m)[x, θ] generated by a monic central element f Z/(p s )[X m ] of degree n is a two sided principal ideal. The skew polynomials of degree less than n are canonical representatives of the elements of GR(p s, m)[x, θ]/(f). Any right divisor g of f generates a left principal ideal (g)/(f) in GR(p s, m)[x, θ]/(f). The set of left multiples of g by skew polynomials of degree less than k = n deg(g) are canonical representatives in GR(p s, m)[x, θ]/(f) of the elements of (g)/(f). In particular the ideal (g)/(f) has the structure of a submodule of GR(p s, m) n and the coefficient vectors of the elements of (g)/(f) form a code of length n and of dimension k over GR(p s, m). Proof. Consider the left ideal (f) GR(p s, m)[x, θ]. Since f is monic, the degree of any non zero h = t f (f) is at least the degree of f. Since f is a central element, we also have h = f t and a left division of h by f is of the form h = f q+r. Since r = h f q = h q f (f) is of degree less than f we have r = 0, showing that h = f q. The reverse inclusion is obtained in a similar way, showing that the left and right ideals generated by f coincide. Therefore the left or right ideal (f) is a two sided ideal. 4

5 In GR(p s, m)[x, θ]/(f) an element h can be identified with its unique remainder by left (or right) division by f. Therefore the skew polynomials of degree less than n, corresponding to the possible remainders, are canonical representatives of the elements of GR(p s, m)[x, θ]/(f). For a right divisor g of f, the ideal (f) is contained in the left ideal (g). By the correspondence of left ideals we have that (g)/(f) is a left ideal in GR(p s, m)[x, θ]/(f). With the above choice of skew polynomials of degree less than n as canonical representants of the elements of GR(p s, m)[x, θ]/(f), the elements of (g)/(f) are left multiples of g by skew polynomials of degree less than k = n deg(g). The claim now follows. Note that not all left ideals in GR(p s, m)[x, θ]/(f) are principal, but in the following we will focus on those ideals. Definition 2.2 A θ-principal code over GR(p s, m) is the set of coefficient vectors of the code corresponding to an ideal (g)/(f) where f Z/(p s )[X m ] is a monic central polynomial and g a monic right divisor of f. A θ-cyclic code over GR(p s, m) is a θ-principal code over GR(p s, m) where f is of the form X n 1 with m a divisor of n. A θ-constacyclic code over GR(p s, m) is a θ-principal code over GR(p s, m) where f is of the form X n α for α Z/(p s ) and m a divisor of n. A θ-constacyclic code C over GR(p s, m) is a left ideal I GR(p s, m)[x, θ]/(x n α). In particular let (a 0, a 1,..., a n 1 ) C, then a(x) = a 0 +a 1 X +...+a n 1 X n 1 I. Now X a(x) also belongs to I: X a(x) = X ( a 0 + a 1 X a n 1 X n 1) = θ(a 0 )X + θ(a 1 )X θ(a n 2 )X n 1 + θ(a n 1 )X n = α θ(a n 1 ) + θ(a 0 )X + θ(a 1 )X θ(a n 2 )X n 1. Therefore (α θ(a n 1 ), θ(a 0 ), θ(a 1 ),..., θ(a n 2 )) C. For θ-cyclic codes we have α = 1 and we obtain the classical property of cyclic codes when θ is the identity, which justifies the above terminology. If g = X n k + g n k 1 X n k g 1 X + g 0 GR(p s, m)[x, θ] divides on the right a polynomial f Z/(p s )[X m ] of degree n, then the generator matrix of the θ-principal code [n, k] generated by g is given by g 0... g n k θ(g 0 )... θ(g n k 1 ) G = θ k 1 (g 0 )... θ k 1 (g n k 1 ) 1 Note that instead of monic polynomials we could also consider polynomials with invertible leading coefficient. However the set of codes obtained by the above construction would be the same. Example. We have seen that X 4 1 GR(4, 2)[X, θ] has two (not unique) factorizations into irreducible monic polynomials : (X + 1) (X + 1) (X + 2ξ + 1) (X + 2ξ + 3) and (X 2 + 2ξ + 1) (X 2 + 2ξ + 3). One can check that this polynomial has 12 monic factors on the right of degree 1; 40 monic factors on the right of degree 2 and 12 monic factors on the right of degree 3, which gives 64 non trivial θ-cyclic codes of length 4 defined by the corresponding right factors. 5

6 3 The length of a θ-principal code This section is a generalization of [6], Section 2. We will show that any monic skew polynomial g GR(p s, m)[x, θ] divides a central polynomial f Z/(p s )[X m ] generator a two sided ideal and therefore is the generator polynomial of some θ-principal code. The degree N of the central monic polynomial f Z/(p s )[X m ] of smallest degree that g divides is the minimum number of rows that the previous generator matrix has to contain in order for the resulting code to be a θ-principal code over GR(p s, m), i.e. for the corresponding code to have the structure of a principal ideal in the quotient of GR(p s, m)[x, θ] by a principal ideal generated by a unitary central polynomial. Therefore N is a lower bound for the length a θ-principal code over GR(p s, m) generated by g. Definition 3.1 (cf [16]) An element P GR(p s, m)[x, θ] is bounded if the left ideal (P ) contains a two sided ideal (P ). In this case P is a bound for P. We adapt the proof of Theorem 15 in [16]. Since GR(p s, m) = Z/(p s )[ξ],the ring GR(p s, m) is a free Z/(p s ) module of dimension m. Proposition 3.2 If P GR(p s, m)[x, θ] is of degree n, then there exists a bound P for P of degree at most m 2 n. Proof. The elements in GR(p s, m)[x, θ] of degree less than n form a GR(p s, m) module of dimension n and therefore a free Z/(p s ) module of dimension m n. Considering the remainders of the division X m i = Q i P + R i, i = 0, 1,..., m n, with deg(r i ) < n, there exists a non trivial linear combination m n δ ir i = 0 where δ i Z/(p s ). This shows that ( m n m n ) δ i X m i = δ i Q i P. The above polynomial m n δ ix m i is a bound for P. This degree bound can be improved in the special case of GR(4, 2). Lemma 3.3 If P GR(4, 2)[X, θ] is of degree n, then there exists a bound P for P of degree at most 2n. Proof. Write P = n p ix i. Define ˆP = n ( 1)i+1 θ i+1 (p i )X i. Let Q := P ˆP then for l {0,..., 2n}, the coefficient of degree l of Q is q l = p i ( 1) j+1 θ l+1 (p j ) i+j=l As θ is of order 2, if l is odd then q l = i+j=l p i( 1) j+1 p j = i even p ip l i i odd p ip l i = 0. If l is even then q l = i+j=l p i ( 1) l i+1 θ(p j ) so θ(q l ) = i+j=l θ(p i) ( 1) i+1 p j = q l. To conclude, q l cancels if l is odd and is fixed by θ if l is even, so the polynomial Q is in Z/(4)[X 2 ] which proves that it is a bound for P. As the bound of a polynomial g GR(4, 2)[X, θ] is at most of degree 2 deg(g), such a polynomial will always generate a θ-principal code of length 2deg(g). Since only the degree and not the explicit knowledge of the bound g is not needed in the generator matrix of the θ-principal code (g)/(g ), it is easy to compute all codes of length n for generator polynomials of degree n k at most n/2. We will use the following mapping from GR(4, 2) to Z 4 (see [14] or section 6): 6

7 1. To each line of the generator matrix of the code over GR(4, 2), we add a line whose entries are multiplied by ξ. 2. The entry a + ξb is replaced by the two entries 3a and a + b. Example. In GR(4, 2)[X, θ] the polynomial X 2 + ξ X + ξ + 1 is a right divisor of the central polynomial X We obtain a θ-constacyclic code whose generator matrix is ( ) ξ + 1 ξ ξ 3ξ The first step of the above transformation gives and the second step gives ξ + 1 ξ ξ + 3 ξ 0 0 3ξ 3ξ ξ ξ which is the generator matrix of the code over Z 4. The following are the best codes over Z 4 obtained this way, compared to the best known codes. In order to obtain a binary code from a code over Z 4 we are using the Gray map (cf. [7]), which is a weight- and distance-preserving map from Z n 4 (with Lee weight metric) to Z2n2 (with Hamming weight metric). For each n and k we compute all [n, k] θ-principal codes over GR(4, 2) and find the best minimal Lee weight d of these codes over Z 4. In the table, N = 4n is the length and 2 K = 2 4k is the size of the resulting binary code, A(N, d) is the size of the largest binary code of length N and Hamming distance d obtained from the table in [10]. N \ K d C d C d C d C best : A(7, 3) = best : A(15, 7) = 2 5 A(15, 5) = best : A(23, 11) = A(23, 7) = 2 12 A(23, 5) = 2 14 A(23, 3) = N \ K d C d C d C d C best : A(31, 7) = 2 17 A(33, 5) 2 22 A(31, 3) = A(41, 9) best : 2 20 A(63, 5) A 3 39 = best : A(47, 9) = A(63, 7)

8 4 Parity check matrix and Euclidean duals of θ-constacyclic codes In this section we extend the results of [6] on self-dual skew cyclic codes. A code over GR(p s, m) is Euclidean self-dual if it is equal to its dual w.r.t. the form < x, y >= i x i y i We shall prove that, if α 2 = 1 and if m divides n, then the Euclidean dual of a θ-constacyclic code (g)/(x n α) GR(p s, m)[x, θ]/(x n α) is again a θ-constacyclic code (g )/(X n α) GR(p s, m)[x, θ]/(x n α). The following lemma explains why the two factors in the decomposition of the generator of a central monic polynomial in two monic polynomials always commute: Lemma 4.1 Suppose that f Z/(p s )[X m ] is a monic polynomial which decomposes into a product of monic polynomials as h g over GR(p s, m)[x, θ], then h g = g h in GR(p s, m)[x, θ]. Proof. Since h g is a central element we have (h g) h = h (h g). Therefore h (g h h g) = 0. Since the leading coefficient of h is invertible, h is not a zero divisor, showing that h g = g h in GR(p s, m)[x, θ]. Using this commutativity result, we can proceed as in the cyclic case to obtain a parity check polynomial: Lemma 4.2 Suppose that f Z/(p s )[X m ] is a monic polynomial which decomposes into a product of monic polynomials as f = h g over GR(p s, m)[x, θ] and denote by C the θ-principal code corresponding to the left ideal generated by g in GR(p s, m)[x, θ]/(f). Then a C if and only if a(x) h = 0 in GR(p s, m)[x, θ]/(f). Proof. If a C, then a(x) = u g. By the above commutativity result we get a(x) h = (u g) h = u (h g) = 0 in GR(p s, m)[x, θ]/(f). Conversely, if a(x) h = 0 in GR(p s, m)[x, θ]/(f), then a(x) h = u f = u (h g) = (u g) h in GR(p s, m)[x, θ]. Like in the above proof we use the fact that h is not a zero divisor to obtain a(x) = u g, showing that a C. The parity check matrix is now obtained from the condition a C a(x) h = 0 in GR(p s, m)[x, θ]/(f) : Lemma 4.3 Suppose that m divides n, that α Z/(p s ) and that X n α Z/(p s )[X m ] decomposes as a product of unitary polynomials X n α = h g over GR(p s, m)[x, θ]. Let C be the θ-constacyclic code corresponding to the left ideal generated by g in GR(p s, m)[x, θ]/(x n α). If g = n k 1 g i X i + X n k and h = k 1 h ix i + X k, then the following matrix 1... θ k 1 (h 1 ) θ k (h 0 ) θ k+1 (h 0 ) θ n 2 (h 1 ) θ n 1 (h 0 ) 8

9 is a parity check matrix for C. Proof. Lemma 4.2 shows that for a(x) C the product a(x) h = 0 in GR(p s, m)[x, θ]/(x n α). Now deg(a(x) h) < n + k and from this we deduce that the coefficients of the monomials X k, X k+1,..., X n 1 in this product must be zero. As, for l {k,..., n 1}, the coefficient of X l in a(x) h is k a l j θ l j (h j ) j=0 we get the result. Comparing the rank of the above matrik and the dimension of the dual code, we can now conclude. Theorem 4.4 Suppose that m divides n, that α Z/(p s ) is such that α 2 = 1 and X n α Z/(p s )[X m ] decomposes as h g over GR(p s, m)[x, θ]. Denote by g = n k 1 g i X i + X n k and h = k 1 h ix i + X k. The Euclidean dual of the θ-constacyclic code (g)/(x n α) is the θ-constacyclic code (g )/(X n α), where g = 1 + θ(h k 1 )X θ k (h 0 )X k. Proof. According to the previous result we need to show that the above matrix H is the matrix of a θ-constacyclic code, which amounts to show that θ k (h 0 )X k θ(h k 1 )X + 1 is also a right divisor of X n α. The ring GR(p s, m)[x, θ] can be localized to the right at the multiplicative set S generated by X consisting of all integer powers of X n 1 where n 1 > 0. This follows from [22] Theorem 2 (see also [8] p. 162) since S verifies the following two necessary and sufficient conditions 1. Condition 1 (right Ore condition): for all X n 1 S and f 1 GR(p s, m)[x, θ], there exists X n 2 S and f 2 GR(p s, m)[x, θ] such that f 1 X n 1 = X n2 f 2. To prove this we note that the multiplication rule X n 1 a = θ n 1 (a)x n 1 allows to shift powers of X from left to right by changing the coefficients. 2. Condition 2: if for X n 1 S and f 1 GR(p s, m)[x, θ] we have X n1 f 1 = 0, then there exists X n 2 S such that f 1 X n 2 = 0. But since X n 1 is never a zero divisor, f 1 must be zero. This shows that the right localization GR(p s, m)[x, θ]s 1 exists. We have ax 1 = X 1 θ(a) where X 1 is the inverse of X in this ring. We now consider the ring R GR(p s, m)[x, θ]s 1 consisting of the elements n X i a i, where the coefficients are on the right and where the multiplication rule is given by ax 1 = X 1 θ(a). The ring R is isomorphic to the skew polynomial ring GR(p s, m)[x 1, θ 1 ]. The map ϕ: GR(p s, m)[x, θ] R GR(p s, m)[x, θ]s 1 n a i X i n X i a i is an anti-isomorphism of rings. For P 1 = r a ix i and P 2 = t b ix i we have ϕ(p 1 + P 2 ) = ϕ(p 1 ) + ϕ(p 2 ) and r+t ϕ(p 1 P 2 ) = ϕ ( a i θ i (b j ))X k = X k a i θ i (b j ) k=0 i+j=k k i,j = X j X i θ i (b j )a i = X j b j X i a i k i,j = ϕ(p 2 )ϕ(p 1 ) 9 k i,j

10 If X n α = g h, then we have X k ϕ(h) ϕ(g) X n k = X k ϕ(gh) X n k = X k ϕ(x n α) X n k = X k (X n α) X n k = 1 αx n = α ( X n 1 α). If 1 α = α, then Xk ϕ(h) = 1 +θ(h k 1 )X +...+θ k (h 0 )X k = g. Since X k ϕ(h) and ϕ(g) X n k both belong to GR(p s, m)[x, θ], we get that g divides X n α in GR(p s, m)[x, θ]. 5 Euclidean Self-dual θ-constacyclic Codes over GR(4, 2) Our goal in this section is to compute all Euclidean self-dual θ-constacyclic codes of length n 20 over GR(4, 2)[X, θ]. We recall (section 1) that θ is the generalized Frobenius automorphism defined by θ(a + 2b) = a 2 + 2b 2 for a, b T = {0, 1, ξ, ξ 2 } and ξ 2 + ξ + 1 = 0. The approach is a generalization of [6]. We need to find all skew polynomials g which divide X n α in GR(4, 2)[X, θ] with α {1, 3} and such that the θ-constacyclic code C = (g)/(x n α) is Euclidean self-dual. This implies that n = 2k where k is both the dimensions of C and C. Theorem 4.4 allows to express the coefficients of the generator polynomial g of C in terms of the coefficients of the polynomial h defined by g h = h g = X 2k α. For C to be Euclidean self-dual, g and g must differ by a constant multiple. This allows to express the coefficients of h in terms of the coefficients of g. Equating the coefficients of X 2k α (g g ) to zero, produces a (commutative) polynomial system of equations over GR(4, 2) for the coefficients of all skew polynomials g for which C = C. All possible generators g of Euclidean self-dual θ-constacyclic codes of given length can then be determined by computing a Groebner basis for this polynomial system. Let us now detail the computation of the coefficients of the generator polynomial for an Euclidean self-dual θ-constacyclic code. Let g = k g ix i with g k = 1 the monic generator polynomial of an Euclidean self-dual θ- constacyclic code over GR(4, 2). Let h = k h ix i with h k = 1 such that X 2k α (g h) = 0. From theorem 4.4, we get g = k θi (h k i )X i and as C is Euclidean self-dual, we must have g = θ k (h 0 ) k g ix i. From these two expressions of g we deduce the coefficients of h: h = k θ k i (g k i g 1 0 )Xi So the monic polynomials g of Euclidean self-dual θ-constacyclic codes of length 2k are characterized by the relation ( k 1 ) ( ) k g i X i + X k θ k (g0 1 ) + θ k i (g0 1 g k i )X i = X 2k α i=1 In Magma, we write the coefficients g i in their 2-adic representation as a i + νb i where ν is an indeterminate representing 2. So we define the polynomial ring F 4 [a 0,..., a k 1, b 0,..., b k 1 ][ν] and consider the relations a 4 i = a i, b 4 i = b i, ν 2 = 0 (1) Then g 1 0 = a 2 0 ν a 0 b 0 10

11 We use the addition rules of 2-adic numbers ([23], Section 6.2) and for α {1, 1 + ν}, we compute the coefficients of g h (X 2k α). They are of the form P (a i, b j ) + νq(a i, b j ) where P, Q F 4 [a 0,..., a k 1, b 0,..., b k 1 ] and must cancel. So each coefficient leads to two polynomial relations P (a i, b j ) = Q(a i, b j ) = 0 (2) We compute a Groebner basis for all the algebraic relations (1) and (2) over F 4. We get the coefficients g i in their 2-adic representation and convert them in their additive representation i.e. in Z 4 [ξ]. We made computations for n = 2k 20 and α {1, 3} and get Euclidean self-dual θ- constacyclic codes only for n {4, 12, 20} and α = 3. In the next section we give the genrator polynomials that we obtained and following [14], we deduce from them self-dual codes over Z 4. For each (monic) generator polynomial g of an Euclidean self-dual code of length n we construct the corresponding generator matrix (given in section 2) over GR(4, 2) from which we deduce the generator matrix of a self-dual code over Z 4 of length 2n (using the mapping of [14] already given in section 3 and explained in the next section). 6 Self-dual Z 4 -codes For any missing definition on Z 4 codes (resp. lattices) we refer to [23] (resp. [12]). A lattice of dimension n is a discrete additive subgroup of R n of maximal Z rank. The dual L of a lattice L is given by L := {x R n : y L, < x, y > Z}, where < x, y > stands for the standard inner product of x, y R n. The norm µ of a lattice is the quantity µ := min{< x, x >: x 0 & x L}. A lattice L is unimodular iff L = L. It is then Type II if x L, < x, x > 0 (mod 2), and Type I otherwise. A lattice is l modular for some prime l if L is similar to L/ l. It was proved in [21, Th. 1] that the norm of a unimodular n dimensional lattice is at most µ 2( n/24 + 1), for n 23. A unimodular lattice meeting that bound is called extremal. Similarly, it was proved in [21, Th. 2] that the norm of a 3 modular n dimensional lattice is at most µ 2( n/12 + 1), for n even. A 3 modular lattice meeting that bound is called extremal. For more details and motivation see [12, 21]. A linear code of length n over Z 4 is a submodule of Z n 4. The dual C is understood with respect to the standard inner product. A code is self-dual if it is equal to its dual. It is then Type IV if, furthermore, the Hamming weight of each of its codewords is even. 11

12 The Euclidean weight of a vector x = (x 1, x 2,..., x n ) is : w E (x) := n min{x 2 i, (4 x i ) 2 }. i=1 The Lee weight of a vector x = (x 1, x 2,..., x n ) is : w L (x) := n min{ x i, (4 x i ) }. i=1 The composition of a vector x Z n 4 say n i(x) for i = 0, 1, 2 is the number of entries= ±i in x. For instance w L (x) = n 1 (x) + 2n 2 (x). The symmetrized weight enumerator (swe) of a code C is then defined as swe C (a, b, c) = x C a n 0(x) b n 1(x) c n 2(x). The Euclidean weight enumerator (ewe) of a code C is then defined as ewe C (a, b) = x C a 4n w E(x) b w E(x). A self-dual code is Type II if all vectors in the code have Euclidean weights which are 0 (mod 8) and Type I otherwise. The minimum Euclidean (resp. Lee) weight of the code is denoted by d E (resp.d L ). We shall recall the standard A 4 construction of a lattice from a self-dual code over Z 4. Define the reduction modulo 4, by ρ : Z n Z n 4, by ρ(x 1,..., x n ) = (x 1 mod 4,..., x n mod 4). Given a code C over Z 4 we construct a lattice by Λ(C) = 1 2 {x Zn ρ(x) C}. (3) It is shown in [4] that if C is a Type I code then Λ(C) is a Type I unimodular lattice, and that if C is a Type II code then Λ(C) is a Type II unimodular lattice and that the minimum norm of the lattice is min{4, d E 4 }. For a notion of Type II codes over GR(4, 2) we refer to [2]. From Euclidean self-dual codes over GR(4, 2), one can construct self-dual codes over Z 4 by projection on the trace orthogonal basis (ξ + 3, ξ) (see theorem 4.1 of [14]). Namely, if C is an Euclidean self-dual code over GR(4, 2) then ν(c) is a self-dual code over Z 4 where ν is the mapping defined in section 4.1 of [14] by : ν(c (ξ + 3) + d ξ) = (c, d) for c, d Z 4, i.e. ν(a + bξ) = (3a, a + b) for a, b Z 4. For n in {4, 12, 20}, we give the codes of length n found via the Groebner basis computation explained in section 5. After projection over Z 4, we classify them in classes of self-dual codes over Z 4 with same symmetric weight enumerator (swe) and Euclidean weight enumerator (ewe). The weight enumerators that are not displayed can be obtained from the companion research report. For each class, we give the minimum Euclidean weight d E, the minimum Hamming distance d, the minimum Lee weight d L, a generator polynomial g, and the number of codes of the class. For codes whose Euclidean distance is 8 and which are of Type II, we compute the root systems generated by the short vectors of their lattices. 12

13 d E d d L Generator swe & ewe polynomial g swe = a a 4 b a 4 c a 3 b 4 c X a 2 b 4 c ab 4 c b b 4 c 4 + c 8 (3 ξ + 1) X + ξ ewe = a a 28 b a 24 b a 20 b a 16 b a 12 b 20 + b 32 swe = a a 4 c a 3 b X 2 + c a b 4 c b 8 + c 8 (3 ξ + 3) X + 3 ξ ewe = a a 24 b a 16 b 16 + b 32 Table 1: Euclidean Self-dual θ-constacyclic Codes (g)/(x 4 + 1) For n = 4, we find 8 codes which are classified in two classes of 4 codes (table 1). The second code is a Type I code called E 8 in [13]. For n = 12, we get 28 classes of self-dual codes of length 24 over Z 4 with the same symmetric weight enumerator and Euclidean weight enumerator. We classify them according to their Euclidean distance d E {4, 8, 12, 16}. There are only 4 codes (with same ewe and swe) with Euclidean distance d E = 4. Their distances are d = 4 and d L = 4. One of the generator polynomials of this unique class is g = X 6 + (3 ξ + 1) X 3 + ξ The codes with Euclidean distance d E = 8 are classified according their type. There are 8 classes of Type I codes and 12 classes of Type II codes. The 156 Type I lattices (a.k.a. odd unimodular lattices) in dimension 24 are uniquely characterized by their root systems formed by their norm 2 vectors [12, Chap. 17]. These are indicated below (table 2) as among A 24 1, A8 2, A8 3, D6 4. So 8 distinct swe s only yield 4 distinct lattices. It is an open and challenging problem to recover all 156 Type I lattices by Construction A 4 as it has been done for the 24 Niemeier lattices in [3]. There are exactly 23 unimodular even lattices of norm 2 in dimension 24. They were classified by Niemeier and later by Venkov [12, chap. 18], and are uniquely characterized by the root systems spanned by their norm 2 vectors. We compute the systems of roots of the lattices obtained by the Type II codes by Construction A and find A 24 1, A8 3, D6 4, D4 6, D2 12 and E3 8 (table 3). The codes of length 24 over Z 4 and Euclidean distance d E = 12 (table 4) are of Type I and give by Construction A the so-called Odd Leech lattice, the unique unimodular lattice of norm 3 in dimension 24 [12, Chap. 17]. They are distinct from the four codes in [15] as their swe s are different (inspection of the monomial terms in a 12 c 12 and a 15 b 8 c). The Type II codes of length 24 and Euclidean distance d E = 16 (table 5) give rise to the Leech lattice by Construction A. Since their Lee weight is only 8 (and not 12) they are not one of the thirteen Lee-optimal codes classified by Rains [20]. 13

14 Root d d L Generator polynomial of syst. g codes A X ξx X 4 + (3 ξ + 1) X ξx X + ξ X X 5 + (ξ + 3) X 4 + (2 ξ + 1) X (3 ξ + 2) X 2 + X + 1 A X 6 + X 5 + (3 ξ + 2) X ξx (ξ + 2) X 2 + (ξ + 1) X + 3 ξ X 6 + (2 ξ + 1) X 5 + (3 ξ + 1) X 4 + (2 ξ + 1) X (ξ + 2) X 2 + (2 ξ + 1) X X 6 + (ξ + 3) X 5 + (3 ξ + 3) X 4 + ξx 2 + (ξ + 2) X A X 6 + (ξ + 1) X ξx X 3 + (ξ + 1) X 2 + ξ X X 6 + (ξ + 3) X 5 + ξx 4 + (3 ξ + 3) X 2 + (ξ + 2) X D X X X 4 + (3 ξ + 1) X ξx ξx + ξ 4 Table 2: Type I Euclidean Self-dual θ-constacyclic Codes (g)/(x ) with d E = 8 Lastly, for n = 20, we did not compute the swe and ewe of all codes. by the polynomial The one generated g = X X 9 + (2ξ + 1) X 8 + (ξ + 3) X 6 + (2ξ + 1) X 5 + (3ξ + 2) X 4 + (2ξ + 3) X X + 1 has mimimum Hamming distance d = 8, minimum Euclidean distance d E = 16 and minimum Lee distance d L = 14. It is therefore better for the Lee and Euclidean distance than the best possible Type IV code in length 40 [9]. 14

15 Root d d L Generator polynomial of syst. g codes 4 8 X 6 + X 5 + (ξ + 3) X 4 + (ξ + 3) X 2 + ξ X + 3 ξ 8 A X 6 + X 5 + (ξ + 3) X 4 + 3X 3 + (ξ + 2) X 2 + X X 6 + (3ξ + 3) X 5 + (ξ + 1) X 4 + ξx 2 + ξx X 6 + (3 ξ + 3) X 5 + (3 ξ + 2) X A X 3 + (3 ξ + 1) X 2 + ξ X + 3 D 6 4 D 4 6 D X 6 + (2 ξ + 2) X 5 + (3 ξ + 3) X 3 + (2 ξ + 2) X + 3 ξ X 6 + (3 ξ + 1) X 5 + (3 ξ + 1) X (3 ξ + 2) X 2 + (ξ + 2) X X 6 + (3 ξ + 1) X 5 + ξx X 3 + (ξ + 1) X 2 + (ξ + 2) X X 6 + (2 ξ + 3) X 5 + (3 ξ + 1) X X 3 + (3 ξ + 2) X 2 + (2 ξ + 1) X X 6 + (3 ξ + 1) X 5 + (ξ + 2) X (ξ + 3) X 2 + (ξ + 2) X X 6 + X 5 + (ξ + 3) X 4 + X 3 + (ξ + 2) X 2 + X X ξx X 4 + (3 ξ + 3) X ξx X + 3 ξ 4 E X 6 + (3ξ + 3)X 3 + 3ξ 4 Table 3: Type II Euclidean Self-dual θ-constacyclic Codes (g)/(x ) with d E = 8 15

16 d d L Generator polynomial swe of g codes 4 8 X 6 + (ξ + 1) X 5 + (ξ + 1) X 4 + a a 15 b 8 c X ξx 2 + ξ X a 12 c X 6 + (ξ + 1) X 5 + (3 ξ + 2) X 4 + a a 15 b 8 c+ 4 (ξ + 3) X 2 + ξx a 12 c X 6 + (2 ξ + 2) X 5 + a a 15 b 8 c+ 4 (3 ξ + 1) X 3 + (2 ξ + 2) X + ξ a 12 c X X 5 + (ξ + 3) X 4 + a a 15 b 8 c+ 16 (2 ξ + 3) X 3 + (3 ξ + 2) X 2 + X a 12 c 12 + Table 4: Euclidean Self-dual θ-constacyclic Codes (g)/(x ) with d E = 12 d d L Generator polynomial of g codes 4 8 X 6 + (3 ξ + 3) X 5 + (ξ + 3) X X 3 + (ξ + 2) X 2 + ξx X 6 + (3 ξ + 3) X ξx 4 + (3 ξ + 3) X 2 + ξx X X X 4 + (3 ξ + 3) X ξx ξ X + 3 ξ 4 Table 5: Euclidean Self-dual θ-constacyclic Codes (g)/(x ) with d E = 16 7 Hermitian Self-dual θ-constacyclic Codes over GR(4, 2) We compute Hermitian self-dual θ-constacyclic codes over GR(4, 2), which means Hermitian self-dual codes generated by polynomial which divide f = X n α with α {1, 3}. We use the same techniques as for Euclidean self-dual codes with the scalar product < x, y > H = n x i θ(y i ) i=1 Following lemma 21 of [6] and theorem 4.4, we get Proposition 7.1 Suppose that n is an even integer and let k N. Let g and h = k 1 h i X i + X k be elements of GR(4, 2)[X, θ] such that h g = g h = X n α where α {1, 3}. The Hermitian dual of the θ-constacyclic code (g)/(x n α) is the θ-constacyclic code (g H )/(X n α) where g H = 1 + k θ i+1 (h k i ) X i i=1 Proof. Let g H = 1 + k i=1 θi+1 (h k i ) X i. First, we check that the polynomial g H generates a θ-constacyclic code. Namely, let g the generator polynomial of the Euclidean dual of the code (g)/(x n α). Then g H = φ(g ) where φ is the morphism from GR(4, 2)[X, θ] to GR(4, 2)[X, θ] defined by φ( a i X i ) = θ(a i )X i. As g is a right divisor of X n α, g H = φ(g ) is also a right divisor of φ(x n α) = X n α so it generates a θ-constacyclic code. Now, let c(x) (g)/(x n α), then for i in {0,... n k 1}, < c(x), X i g H > H =< c(x), X i g >= 0 16

17 where < a(x), b(x) > H =< a, b > H and < a(x), b(x) >=< a, b >. So the Hermitian dual of (g)/(x n α) is (g H )/(X n α) In GR(4, 2)[X, θ], the polynomial h of a Hermitian self-dual code (g)/(x 2k α) defined by h g = g h = X 2k α becomes k h = θ k+1 (g0 1 ) + θ k i+1 (g0 1 g k i )X i i=1 so the generators g of Hermitian self-dual θ-constacyclic codes of length 2k are characterized by the relation ( k 1 ) ) k 1 g i X i + X (X k k + θ k i+1 (g0 1 g k i)x i + θ k+1 (g0 1 ) = X 2k α i=1 We made the computations for k 10 and found Hermitian self-dual codes for each k and for α = ( 1) k mod 2. According to [14], from an Hermitian self-dual code of length n, one can construct a 3-modular Z-lattice of dimension 2n whose norm is bounded by 2 n Following [14], the Gram matrix M of the lattice is obtained from the generator matrix of the code as M = 1 U t U + V t V 1 2 U t V 1 2 V t U 1 2 U t U 1 2 V t V 1 2 U t V + V t U U t U 1 2 V t V + U t V 1 2 V t U U t U + V t V 1 2 U t V 1 2 V t U ( where U = G 0 ) ( G1, V = 0 0 ) and G 0 + ξ G 1 is the generator matrix of the code. 0 4I N We compare our results with the table of best lattices, given in [18]. In the first row, we give the length n = 2k; in the second row, the generator polynomial of a Hermitian self-dual code C of length n, (g)/(x 2k α) where α = ( 1) k mod 2 ; in the third column, the norm of the lattice constructed from the code C and in the last column, the best known norm (BKN) for 3-modular Z-lattices of dimension 2n. We notice that the 36 lattices of norm 6 we obtain in length n = 14 are all isometric to one of the two known extremal lattices of dimension 28, Beis 14 in the notation of [18]. This lattice was previously constructed by combining Construction A modulo 2 over the Eisenstein integers with Kneser neighboring [1]. We give here another construction of this lattice, without taking neighbors. Similar trade off between alphabet size and neighboring can be observed in [11]. 8 Cubic Construction Following [14], from self-dual codes over Z 4 of length l and Hermitian self-dual codes of length l over GR(4, 2), one can construct 3l self-dual codes overs Z 4. We construct self-dual codes over Z 4 with length 24 and get codes of Euclidean weight 8, 12 or 16. We focus on the Type II codes of Euclidean weight 8 and get the following systems of roots, which improves the results obtained in the previous section : A 12 2, A8 3, A4 6, A3 8, A 24, D4 6, D4 6, D3 8, D12 2 and D 24. More precisely, for each of the 7 self-dual codes C 1 of length 8 over Z 4 ([13]) and each of the 16 Hermitian self-dual codes C 2 with length 8 over GR(4, 2) previously computed, we construct self-dual codes over Z 4 of length 24. If their minimum Euclidean weight is 8 and if the codes are 17

18 Length Generator polynomial Norm BKN 2k g 4 X ξ X X X + 2 ξ X X X + 2 ξ X X X ξ X X X ξ X 7 + (3 ξ + 1) X 6 + (ξ + 2) X 5 + (ξ + 1) X 4 + (3 ξ + 2) X (3 ξ + 3) X 2 + ξ X + 2ξ X X X ξ X X 7 + (3 ξ + 1) X X X 4 + ξ X X ξ X 10 + (2 ξ + 1) X 8 + (3 ξ + 2) X X 5 + ξ X 4 + X ξ + 1 Table 6: Hermitian Self-dual θ-constacyclic Codes (g)/(x 2k α) with α = ( 1) k mod 2 of Type II, we compute their root system. In first column of the table, we give the root system; in the second column, the generator matrix of C 1 (G 1, G 2, G 3 or G 4 ); in the third column, the generator polynomial of C 2 ; in the last column the number of codes which have the same symmetric weight enumerator and Euclidean weight enumerator that the code constructed from C 1 and C 2. There is only one class of codes for each of the root systems A 12 2, A4 6, A3 8, A 24, D6 4 and D12 2. There are two classes of codes with root system A8 3, D4 6 or D3 8. Root system Self-dual code Hermitian self-dual code of C 1 over Z 4 C 2 over GR(4, 2) equiv. codes A 12 2 G 1 X X X + 2 ξ A 8 3 G 2 X X ξ G 1 X ξ A 4 6 G 1 X X ξ A 3 8 G 3 X X X + 2 ξ A 24 G 3 X X ξ D4 6 G 3 X ξ G 2 X X X X + 2 ξ D6 4 G 2 X ξ D8 3 G 4 X X X + 2 ξ G 4 X X X X + 2 ξ D12 2 G 3 X ξ D 24 G 4 X ξ G 4 X X ξ Table 7: Self-dual Codes over Z 4 with length 24 obtained from cubic construction 18

19 G 1 = G 3 = Open Problems, G 2 = and G 4 = There are several possible generalizations to this approach: 1. In the situations of the previous section we defined codes using the factor rings GR(p s, m)[x, θ]/(f) where f is a monic central polynomial. Any monic right factor of f generates a principal ideal (g)/(f) GR(p s, m)[x, θ]/(f) and therefore corresponds to a linear code. Those codes have properties similar to that of the classical cyclic codes, in particular any code word is a multiple of g and there will always be (p sm ) deg(f) deg(g) code words over GR(4, 2). We may also consider ideals that are generated by non unitary polynomials whose leading coefficients are not invertible and more generally non principal ideals. 2. We may consider arbitrary two sided ideals of GR(4, 2)[X] instead of the principal ideals generated by a central polynomial f. According to [19] Section XX Proposition XX.3 and Exercise XX.11.c the two sided ideals of GR(4, m)[x, θ] are of the form I = (f, 2 f 1 ) where f and f 1 are monic polynomials. Any left ideal in GR(4, m)[x, θ]/i is a linear code over GR(4, m). Therefore a first generalization would be to analyze the left ideals in those factor rings. Acknowledgement: The authors wish to thank the anonymous referees for helpful remarks that improved the content and presentation of the material. References [1] C. Bachoc, Applications of coding theory to the construction of modular lattices, J. Comb. Th A 78-1 (1997) [2] K. Betsumiya, Y. Choie, Jacobi forms over totally real fields and Type II codes over Galois Rings GR(2 m, f) Europ. J. of Comb., 25 (2004) [3] A. Bonnecaze, P. Gaborit, M. Harada, M. Kitazume and P. Solé, Niemeier lattices and type II codes over Z 4, Discrete Math, 205 (1999), no. 1-3, [4] A. Bonnecaze, A.R. Calderbank and P. Solé, Quaternary Quadratic Residue Codes and Unimodular Lattices, IEEE Trans. on Information Theory IT, 41 (1995) [5] D. Boucher, W. Geiselmann and F. Ulmer, Skew Cyclic Codes, Applied Algebra in Engineering, Communication and Computing, 18 (2007), ,. 19

20 [6] D. Boucher and F. Ulmer, Coding with skew polynomial rings Prépublication IRMAR 08-07, to appear in Journal of Symbolic Computation [7] W. Bosma, J. Cannon and C. Playoust, The Magma Algebra System I: The User Language, Journal of Symbolic Computation, 24 (1997), pp [8] N. Bourbaki, Algèbre commutative, Chapitre II, Paris, Hermann, [9] S. Bouyukleva and M. Harada, On Type IV self-dual codes over Z 4, Discrete Math 247 (2002) [10] A.E. Brouwer, Server for bounds on the minimum distance of q-ary linear codes, q = 2, 3, 4, 5, 7, 8, 9, aeb/, (2005) [11] R. Chapman and P. Solé, Universal Codes and Unimodular Lattices, J. Théorie des Nombres de Bordeaux, (1996) [12] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Third edition, Grundlehren der Mathematischen Wissenschaften 290, Springer-Verlag, New York, 1999 [13] J.H. Conway and N.J.A. Sloane, Self-dual codes over the integers modulo 4, J. of Comb. Th. A, 62, (1993) [14] P. Gaborit, A. M. Natividad and P. Solé, Eisenstein Lattices, Galois Rings and Quaternary Codes, International Journal of Number Theory Volume, 2 (2006), [15] T. A. Gulliver and M. Harada, Certain self-dual codes over Z 4, and the odd Leech lattice, Springer Lect. Not. in Comp. Sc (1997) [16] N. Jacobson, The theory of rings, Publication of the AMS., [17] S. Litsyn, E.M. Rains and N. J. A. Sloane, Table of Nonlinear Binary Codes njas/codes/and/ [18] G. Nebe and N. J. A. Sloane, Table of Highest Minimal Norms of Modular Lattices, njas/lattices/modular.html [19] B.R. McDonald, Finite Rings with Identity, Marcel Dekker Inc., [20] E.M. Rains, Optimal self-dual codes over Z 4, Discr. Math., 203 (1999) [21] E.M. Rains, N.J.A. Sloane, The shadow theory of modular and unimodular lattices, J. of Number Theory, 73 (1998) [22] P. Ribenboim., Sur la localisation des anneaux non commutatifs, Séminaire Dubreil. Algèbre et théorie des nombres, tome 24. ( ). [23] Z-X. Wan, Quaternary Codes, World Scientific, address: delphine.boucher@univ-rennes1.fr address: sole@i3s.unice.fr address: felix.ulmer@univ-rennes1.fr 20