5-SEEDs from the lifted Golay code of length 24 over Z4

. Spontaneous emission error designs (SEEDs) are combinatorial objects that can be used to construct quantum jump codes. The lifted Golay code G 24 of length 24 over Z 4 is cyclic self-dual code. It is known that G 24 yields 5-designs. In this paper, by using the generator matrices of bordered double circulant codes, we obtain 22 mutually disjoint 5-(24 ,k,λ ) designs with ( k,λ ) = (8 , 1) , (10 , 36) , (12 , 1584) and 5-(24 ,k ;22)-SEEDs for k = 8 , 10 , 12 , 13 from G 24 .


Introduction
Let V be a set of n elements and let t, k be integers with 0 < t < k < n. For a collection B of k-subsets (called blocks) of V , (V, B) is called a t-(n, k, λ) design if |{B ∈ B : T ⊂ B}| = λ for all t-subsets T of V . In particular, a t-(n, k, 1) design is called a Steiner system S(t, k, n). A design with no repeated block is called simple. All designs studied in this paper are simple. Two t-(n, k, λ) designs defined over the same set are said to be disjoint if they have no blocks in common. Mutually disjoint simple t-designs can be used to construct t-spontaneous emission error designs (t-SEEDs) (see [1,9,10,18]), and t-SEEDs are combinatorial objects having close connection with quantum jump codes (see [2,11,19]).
Let n, k, t, m be integers with 0 < t < k < n and V be a set of n elements. A t-spontaneous emission error design, denoted by t-(n, k; m)-SEED, is a system B of k-subsets of V with a partition B (1) , B (2) , . . . , B (m) of B satisfying that |{B ∈ B (i) : T ⊂ B}|/|B (i) | = λ T for any 1 ≤ i ≤ m and T ⊂ V with |T | ≤ t, where λ T is a constant depending only on T .
A quaternary code C of length n is a linear block code over Z 4 (the ring of integers modulo 4), i.e., an additive subgroup of Z n 4 . Elements of Z n 4 are termed "vectors" even though Z n 4 is not a vector space, and an element of C is called a codeword. The Hamming weight of a codeword is the number of its non-zero components. The Euclidean weights of the elements 0, 1, 2 and 3 of Z 4 are 0, 1, 4 and 1, respectively. The Euclidean weight of a codeword is just the rational sum of the Euclidean weights of its components.
The symmetrized weight enumerator (or s.w.e.) of a code C over Z 4 is where n 0 (c), n 1 (c) and n 2 (c) are the numbers of components of c equal to 0, ±1 and 2, respectively.
We say that two codes are equivalent if one can be obtained from the other by permuting the coordinates and (if necessary) changing the signs of certain coordinates. Codes differing by only a permutation of coordinates are called permutationequivalent. Equivalent codes have the same s.w.e..
Any quaternary code is permutation-equivalent to a code with generator matrix of the form where A and D are Z 2 -matrices and B is a Z 4 -matrix, and 0 is the k 2 × k 1 zero matrix. A codeword has the form aG, where a = (a 1 , · · · , a k1 , a k1+1 , · · · , a k1+k2 ), We say that a code with generator matrix (1) is of type 4 k1 2 k2 , containing 2 2k1+k2 codewords (cf. [8]). Equation (1) illustrates a difference in point of view between ring theory and coding theory. Quaternary codes are Z 4 -modules. A quaternary code C is not in general a free module [17], and so need not have a basis. The code C is a free Z 4 -modules if and only if k 2 = 0 [13]. In particular, if the code C is free, then the size of the basis is still termed dimension of C.
If a quaternary code C has generator matrix (1), then the dual code C ⊥ has generator matrix where 0 is the k 2 × (n − k 1 − k 2 ) zero matrix, and C ⊥ is of type 4 n−k1−k2 2 k2 . As with codes over finite fields, a generator matrix of C ⊥ is called a parity check matrix of C.
For a ∈ Z 4 , let a denote a vector whose every entry is a. A code with generator matrix of the form is called a bordered double circulant code of length 2s + 2, where I s+1 is the identity matrix of order s + 1 and A is an s × s circulant matrix,.
Throughout this paper, let V n be the set {1, 2, . . . , n}, permutations on a code C of length n are considered on coordinate positions V n of C. For any permutation π acting on C, we denote the image by π(C). However, permutations on C can be considered as column permutations on matrix M whose rows are all codewords in C. But π(C) corresponds to a column permutation on a parity check matrix H of C, that is, if C = {x ∈ Z n 4 : Hx T = 0}, then π(C) = {y ∈ Z n 4 : π(H)y T = 0}, the proof is analogous to the F 2 case in [18].
The paper is organized as follows. In Section 2, we recall background information on cyclic codes over Z 4 and prove several lemmas for later use. In Section 3, we show the intersections among their images of Type II codes over Z 4 with generator matrices of form (3) under some coordinate permutations. In Section 4, we give some constructions of 5-SEEDs from the lifted Golay code G 24 .

Several lemmas
Then it is easy to prove the following formula: To generalize the concept of cyclic codes to codes over Z 4 , we need some algebra.
where the sign in ± is chosen in such a way that the coefficient of the highest power of x is 1. The method of lifting (called, the Hensel lift) a polynomial in Z 2 [x] to a polynomial in Z 4 [x] is known as Graeffe's method [24]. Clearly the inverse mapping is f (x) ≡ F (x)(mod 2). Since both φ and its inverse do not change the degree of a polynomial nor the coefficient of the highest power of x, irreducible polynomials correspond to irreducible polynomials.
The polynomial ring Z 4 [x] is not in general a unique factorization domain. But some special polynomials in Z 4 [x] may have the unique factorization property. An example of such polynomials is x n − 1 if n is odd, and x n − 1 can be factored into unique product of monic basic irreducible pairwise coprime polynomials in where the f i are basic irreducible and pairwise coprime [16,Theorem 12.3.7].
As usual, a cyclic code C of length n over Z 4 is a linear code with the property that if (a 0 , a 1 , . . . , a n−1 ) ∈ C, then (a n−1 , a 0 , a 1 , . . . , a n−2 ) ∈ C. Let R n = Z 4 [x]/(x n −1). Suppose that an element c = (a 0 , a 1 , . . . , a n−1 ) ∈ C is identified with a polynomial a 0 + a 1 x + · · · + a n−1 x n−1 over Z 4 . Under this correspondence, a code is Z 4 cyclic if and only if it is an ideal in the ring R n . In [21] it was shown that any Z 4 cyclic code C of odd length n has generators of the form f ( , and h(x) are unique, monic polynomials. Furthermore, C has type 4 deg g(x) 2 deg h(x) . The proof is referred to Theorem 1 in [21] or Theorem 12.3.13 in [16]. In particular, if h(x) = 1, then C = f (x) , and C has type 4 n−deg f (x) [16,Corollary 12.3.14].
In what follows we consider C as a principal ideal of R n which is generated by an element g(x) and g(x) is a divisor of x n − 1 for odd n. Moreover, we can easily write down a generator matrix G for C = g(x) , and the rows of G correspond to x i g(x) for 0 ≤ i ≤ n − deg g(x) − 1. The following lemma is the same as [20,Theorem 49] except that the field is replaced by a ring, we state it here without proof.
is a divisor of x n − 1 for odd n over Z 4 and the degree of g(x) is n − k, then C = g(x) has 4 k elements, i.e. the dimension C is k.
Let C = g(x) be a cyclic code over Z 4 and g(x) be a divisor of x n − 1 for odd n. Let h(x) = (x n − 1)/g(x) and h ⊥ (x) denote its reciprocal polynomial, then h ⊥ (x) is a monic divisor of x n − 1 as the constant term of h(x) is ±1, and the dual code C ⊥ = h ⊥ (x) . About these facts, we refer the reader to [16,Theorem 12.3.20]. For a quaternary cyclic code, the following lemma is Z 4 analogous to Theorem 53 in [20].
Lemma 2. Let C be a quaternary cyclic code of odd length n with generator polynomial x − 1. Then C has 4 n−1 elements of Z n 4 consisting of all vectors u with f (u) ≡ 0 (mod 4). Moreover, let C = g(x) be a quaternary cyclic code, then f (u) ≡ 0 (mod 4) for any u ∈ C iff x − 1 divides g(x).
Proof. Noting that C = x − 1 and x n − 1 = (x − 1)(1 + x + · · · + x n−1 ) over Z 4 , we have C ⊥ = 1 + x + · · · + x n−1 since 1 + x + · · · + x n−1 is its own reciprocal polynomial. Then C ⊥ has 4 elements consisting of the vectors a of length n with a ∈ Z 4 . Hence, for any u ∈ C we have f (u) ≡ 0 (mod 4) since u and a with a ∈ Z 4 are orthogonal. Conversely, for any u ∈ Z n 4 , if f (u) ≡ 0 (mod 4), then u and a with a ∈ Z 4 are orthogonal. Hence u ∈ C.
Moreover, let C = g(x) be a quaternary cyclic code, then f (u) ≡ 0 (mod 4) for any u ∈ C iff u and a with a ∈ Z 4 are orthogonal for any u ∈ C iff C ⊆ C iff x − 1 divides g(x).
The following lemma is analogous to Lemma 2.2 in [9] and is very useful for the proofs in the next sections. Proof. Suppose that C = g(x) . Then the dimension of C is k as the degree of g(x) is n − k by Lemma 1. Since g(x)|c(x), the i-th row vector x i c(x) of A is a codeword of C for 0 ≤ i ≤ n − 1. We next prove that the first k rows of A, that is, the vectors c(x), xc(x), . . ., x k−1 c(x) are linearly independent. If they were not, then there would be a i ∈ Z 4 , 0 ≤ i ≤ k − 1, so that Because gcd(x n − 1, c(x)) = g(x), we have the equality a 0 + a 1 x + · · · + a k−1 x k−1 ≡ 0 (mod (x n − 1)/g(x)). Note that (x n − 1)/g(x) is monic polynomial of degree k. We then have a i = 0 for 0 ≤ i ≤ k − 1. Hence, the first k rows of A are linearly independent. This means that the degree of g(x) is n − k if and only if rank(A) = k.
The following lemmas are useful for later use.
By the definitions of a t-(n, k, λ) design and a t-(n, k; m)-SEED, we have that B i , 1 ≤ i ≤ m, are mutually disjoint and they constitute a t-(n, k; m)-SEED.

Intersections of codes
In this section we let C (resp. C ) be a cyclic Type II code over Z 4 of length 2s + 2 for odd s with generator matrix G (resp. G ) of form (3) and A (resp. A ) be an s × s circulant matrix in G (resp. G ), and suppose that a(x) (resp. a (x)) denotes the polynomial corresponding to the first row of A (resp. A ). Let δ = (s + 2 s + 3 · · · 2s + 1) be the coordinate permutation acting on V 2s+2 = {1, 2, . . . , 2s + 2}. We shall study the intersections of the images of C and C under all permutations of the form δ i . For any i, j ∈ {0, 1, . . . , s − 1}, define a polynomial depending on C and C as follows: Proof. Since C and C are self-dual codes, we regard G and G as their parity check matrices, respectively. Note that π(G)x T = 0 for any x ∈ π(C), where π is any coordinate permutation acting on V 2s+2 . Suppose that for any i, j ∈ {0, 1, . . . , s−1}, x ∈ δ i (C)∩δ j (C ), then δ i (G)x T = 0 and δ j (G )x T = 0. So we get the homogeneous system of linear equations whose coefficient matrix is For each l (1 ≤ l ≤ s + 1), we add 3 times of the lth row to the (l + s + 1)th row in M , getting  where 0 s+1 is the zero matrix of order s + 1.
Since 3δ i (A) + δ j (A ) is also an s × s circulant matrix and the polynomial corresponding to the first row of it is 3x i a(x) + x j a (x) (mod x s − 1). By Lemma 3, the rank of M is 2s + 1 − d, and hence the dimension of δ i (C) ∩ δ j (C ) is d + 1.
Note that δ i (v) = v and δ i (1) = 1 for any 0 ≤ i ≤ s − 1. Then we have the following results.
Suppose that a(x) (resp. a (x)) denotes the polynomial corresponding to the first row u (resp. u ) of A (resp. A ). Then we have the following result.
Proof. For any i, j ∈ {0, 1, . . . , s − 1}. Since C (resp. C ) is self-dual and contains the all-ones vector. So we have f (u) ≡ f (u ) ≡ 2 (mod 4). Note that the vector corresponding to x i a(x) is a cyclic shift of u. Similarly, the vector corresponding to x j a (x) is a cyclic shift of u . By formula (4), for the vector u 1 corresponding to (3x i a(x) + x j a (x)) (mod x s − 1), f (u 1 ) ≡ 0 (mod 4). Then by Lemma 2,

5-SEEDs from lifted Golay code G 24
The lifted Golay code G 24 over Z 4 is defined in [4] as the extended Hensel lifted quadratic residue code of length 24. G 24 is a Type II code constructed from the cyclic code with generator polynomial x 11 + 2x 10 + 3x 9 + 3x 7 + 3x 6 + 3x 5 + 2x 4 + x + 3, by appending 3 to the last coordinate of the generator vectors. The s.w.e. of G 24 is given in [12]. It is shown that G 24 is equivalent to double circulant Type II code C 1,11 in [12] with the generator matrix of form (3), where A is an 11 × 11 circulant matrix with the first row c=(2, 3, 3, 1, 2, 1, 1, 0, 1, 0, 0), where we call this Type II code C 1 . Moreover, let C 2 with the generator matrix of form (3) be equivalent to C 1 , where the first row of A is (0, 0, 1, 0, 1, 1, 2, 1, 3, 3, 2) obtained in inverse order from c.
For the G 24 , we have that s = 11 and δ = (13 14 · · · 23) is the coordinate permutation acting on V 24 . The notations v, W , and B i , i = 1, 2, 3, 4 are defined as above section. We then have the following lemma.  [6] it is verified that g ij (C, C ) = x − 1. By Proposition 2, the conclusion follows.