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PAPER
Trellis Properties of Product Codes∗ Haibin KAN†a) and Hong SHEN†† , Nonmembers
SUMMARY In this paper, we study trellis properties of the tensor product (product code) of two linear codes, and prove that the tensor product of the lexicographically first bases for two linear codes in minimal span form is exactly the lexicographically first basis for their product code in minimal span form, also the tensor products of characteristic generators of two linear codes are the characteristic generators of their product code. key words: tensor product, trellis, minimal span form, characteristic generator
1.
Introduction
Trellis representations of linear block codes not only illustrate code structure, but also often lead to efficient trellis based decoding algorithms. For any linear block code, there exists a unique, up to isomorphism, minimal conventional trellis. Furthermore, the minimal conventional trellis for any linear code can be easily constructed from its generator matrix or parity check matrix by several methods, such as BCJR, Forney, and Kschischang-Sorokine constructions [9]. Though the theory on conventional trellises is well developed, much less is known about tail-biting trellises. Many examples show that the complexity of a tail-biting trellis can be much lower than the complexity of the best possible conventional trellis. Recently, Koetter and Vardy built some general theory on tail-biting trellis in [3] and [4]. They proved that any linear tail-biting trellis for a linear code can be constructed as the product of elementary tail-biting trellises, just as the reverse procedure of Kschischang-Sorokine construction. So to construct a minimal linear tail-biting trellis for a linear code is reduced to searching for a minimal linear tail-biting trellis among the products of some elementary trellises. It was also proved in [4] that any minimal linear tail-biting trellis for a linear code C can be constructed from its characteristic generators, and that the sum of span matrices of C and its dual C ⊥ was the constant matrix whose Manuscript received December 22, 2003. Manuscript revised April 16, 2004. Final manuscript received September 7, 2004. † The author is with the Department of Computer Science & Engineering, Fudan University, Shanghai, China. He is also with Graduate School and Information Science, Japan Advanced Institute of Science and Technology, Ishikawa-ken, 923-1292 Japan. †† The author is with Graduate School and Information Science, Japan Advanced Institute of Science and Technology, Ishikawaken, 923-1292 Japan. a) E-mail:
[email protected] ∗ This work is supported by NSF of China (No.60472038), Chengguan Plan (No. 20025001007) and Japan Society for Promotion of Science (JSPS) Research Grant (No. 14380139).
elements are all one. A conjecture on the relation between characteristic generators of a linear code C and its dual was posed in [4]. Kan and Shen [2] discussed the conjecture for self-dual codes and cyclic codes. The tensor product C1 ⊗C2 (product code) of two linear codes C1 and C2 is an important construction, which was introduced in [7] and [8]. Wei and Yang [10] considered the relations among the generalized Hamming weights of linear codes C1 , C2 and C1 ⊗C2 . Koetter and Vardy [4] proved that the trellis of the product code C1 ⊗ C2 could be viewed as the generalized trellis product of some trellises for C1 and C2 . In this paper, some relations among the trellis properties of C1 , C2 and C1 ⊗ C2 are discussed. We prove that the tensor product of two lexicographically first bases for C1 and C2 in minimal span form is exactly the lexicographically first basis for C1 ⊗ C2 in minimal span form. Also, the tensor products of all characteristic generators of C1 and C2 are exactly all characteristic generators of C1 ⊗ C2 . 2.
Preliminaries
We adopt the definitions of conventional or tail-biting trellises for block codes appearing in [3] and [4]. For more details, the readers can refer to the two references. An edge-labelled directed graph is a triple (V, E, A), consisting of a set V of vertices, a finite set A called the alphabet, and a set E of ordered triples (v, a, v ), with v, v ∈ V and a ∈ A, called edges. We say that an edge (v, a, v ) begins at v, and ends at v , and has label a. Definition 1: A tail-biting trellis T = (V, E, A) of depth n is an edge-labelled directed graph with the following property: the vertex V can be partitioned as V = V0 ∪ V1 ∪ · · · ∪ Vn−1
(1)
such that every edge in T begins at a vertex of Vi and ends at a vertex of Vi+1 , for some i = 0, 1, · · · , n − 2, or begins at a vertex of Vn−1 and ends at a vertex of V0 . E = E0 ∪ E1 ∪· · ·∪ En−1 , where Ei is the set of all edges beginning at a vertex of Vi . The sets V0 ,V1 ,· · · ,Vn−1 are called the vertex classes of T . The ordered index set I = {0, 1, · · · , n − 1} induced by the partition in (1) is called the time axis for T . The ordered sequence Θ(T ) = (|V0 |, |V1 |, · · · , |Vn−1 |) is called the state profile of T . A conventional trellis T = (V, E, A) of depth n is almost same as a tail-biting trellis except that the vertex V can be
c 2005 The Institute of Electronics, Information and Communication Engineers Copyright
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partitioned as V = V0 ∪ V1 ∪ · · · ∪ Vn and that every edge in T begins at a vertex of Vi and ends at a vertex of Vi+1 , for some i = 0, 1, · · · , n − 1. Generally, we always assume |V0 | = |Vn | = 1 for a conventional trellis. A cycle of length n in a tail-biting trellis T is a closed path in T through n distinct vertices. A tail-biting trellis T is reduced if any vertex and edge belong to at least one cycle. The set of edges labels along a cycle in T is an n-tuple (a0 , a1 , · · · , an−1 ) over the label alphabet A. Postulating that all cycles in T start at a vertex of V0 , every cycle defines a vector (a0 , a1 , · · · , an−1 ) ∈ An , which is called edge-label sequence in T . Let C(T ) denote the set of all edge-label sequences in T . Then C(T ) is called the edge-label code of T . T is a tail-biting trellis for the block code C over A if C(T ) = C. If every vertex in each vertex class Vi , 0 ≤ i ≤ n − 1, is labelled by a sequence of length ιi over A, where ιi ≥ log|A| |Vi | , then this kind of trellis is called a labelled trellis. Here, we require that all vertex labels within the same vertex class are distinct. Let ι = ι0 + ι1 + · · · + ιn−1 . Then every cycle Γ in a labelled tail-biting defines an ordered sequence of length n + ι over A, consisting of the labels of edges and vertices in Γ. We refer to such a sequence as a label sequence in T . Denote by S (T ) the set of all such label sequences, and call S (T ) a label code of T . For tail-biting trellises T = (V, E, A) and T = (V , E , A ), if Θ(T ) ≤ Θ(T ), i.e., |Vi | ≤ |Vi | for all 0 ≤ i ≤ n − 1, (2) we say that T is smaller than T under Θ , and denote it by T Θ T , where V = V0 ∪ V1 ∪ · · · ∪ Vn−1 , V = V0 ∪ V1 ∪ . If there is at least a strict inequality in (2), we say · · · ∪ Vn−1 that T is strictly smaller than T and denote it by T ≺Θ T . T is a minimal tail-biting trellis for a block code C if there is no tail-biting trellis T for C such that T ≺Θ T . All above notions for conventional trellises can be defined in a completely similar way. We always assume that the alphabet set A is a finite field Fq . A labelled trellis T = (V, E, Fq ) is linear over Fq if T is reduced and S (T ) is a linear code over Fq . An unlabelled trellis T is said to be linear if there exists a vertex labelling of T such that the resulting labelled trellis is linear. Definition 2: Let T = (V , E , A) and T = (V , E , A) be two (conventional or tail-biting) trellises with length n. Then the trellis product T = (V, E, A) of T and T is defined as follows: For any i, 0 ≤ i ≤ n, Vi = {(v i , vi )|vi ∈ Vi , vi ∈ Vi }, Ei = { (vi , vi ), ai + ai , (vi+1 , vi+1 ) |(vi , ai , vi+1 ) ∈ Ei , (vi , ai , vi+1 ) ∈ Ei }, n
n−1
i=0
i=0
where V = ∪ Vi and E = ∪ Ei . Denote it by T = T × T . Clearly, |Vi | = |Vi | · |Vi | and |Ei | = |Ei | · |Ei |. If T and T are trellises for linear codes C1 and C2 , respectively, then T = T × T is a trellis for the linear code C1 + C2 .
For i, j ∈ I = {0, 1, · · · , n − 1}, define closed cyclic interval [i, j] as follows {i, i + 1, · · · , j} if i ≤ j, [i, j] = {i, i + 1, · · · , n − 1, 0, · · · j} if i > j. Semi-open cyclic interval (i, j] = [i, j] \ {i}. Let C be any linear code with length n over the finite field Fq , i.e., C ⊆ Fqn . For a nonzero element c = (c0 , c1 , · · · , cn−1 ) ∈ C, the cyclic interval (i, j] is called a span of c if [i, j] contains all nonzero positions of c, and denoted by [c] = (i, j]. For example, if c = (0, 0, 1, 0, 1, 1, 0) ∈ F27 , then (0, 5], (2, 5], (4, 2] are all spans of c. Let (c) denote the smallest nonnegative integer i such that ci 0, and (c) the largest nonnegative integer j such that c j 0. Obviously, [(c), (c)] contains all nonzero positions of c. ((c), (c)] is called the basic span of c. Given c = (c0 , c1 , · · · , cn−1 ) and its span [c] = (i, j], the corresponding elementary labelled trellis T c can be easily constructed [4]. A basis X = {x1 , x2 , · · · , xk } for C is said to be in minimal span form if (x1 ), (x2 ), · · · , (xk ) are distinct and (x1 ), (x2 ), · · · , (xk ) are distinct. F. Kschischang [6] proved the following important result: Lemma 3: Let X = {x1 , x2 , · · · , xk } be a basis for a linear code C. Then T = T x1 × T x2 × · · · × T xk is the minimal conventional trellis for C if and only if the basis {x1 , x2 , · · · , xk } is in minimal span form. The proof of the above lemma was also given in [9]. Given any basis for a linear code C, we can easily get a basis for C in minimal span form by the greedy algorithm. All basic spans of elements in a basis for C in minimal span form are called atomic spans for C. Though a basis for C in minimal span form may be not unique, the atomic spans of any basis for C in minimal span form are uniquely determined by the code C [6], i.e., the atomic spans (xi , xi ], 1 ≤ i ≤ k, are same for any basis {x1 , x2 , · · · , xk } for C in minimal span form. For convenience of notations, we impose the lexicographic order ∝ on the set of vectors in Fqn . Then the lexicographically first basis for C in minimal span form is unique. For c = (c0 , c1 , · · · , cn−1 ) ∈ C, define σi (c) = (ci , · · · , cn−1 , c0 , · · · , ci−1 ), i.e., σi is the cyclic shift to the left i times. So σi (C) = {(ci , · · · , cn−1 , c0 , · · · , ci−1 )| (c0 , c1 , · · · , cn−1 ) ∈ C}, Similarly, for c = (c0 , c1 , · · · , cn−1 ) ∈ C, define ρi (c) = (cn−i , · · · , cn−1 , c0 , c1 , · · · , cn−i−1 ), i.e., the cyclic shift to the right i times. A characteristic generator for C is a pair consisting of a codeword x = (x0 , x1 , · · · , xn−1 ) ∈ C and its span [x] = (a, b] such that xa and xb are nonzero. The set of all the characteristic generators for C is given by X = X0 ∪ X1 ∪ · · · ∪ Xn−1 ∗ = X0∗ ∪ ρ1 (X1∗ ) ∪ · · · ∪ ρn−1 (Xn−1 )
(3)
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with the understanding that [x] = ((x∗ ) + j, (x∗ ) + j] for each x ∈ X j , where X ∗j is the lexicographically first basis for σ j (C) in minimal span form, and x∗ = σ j (x). The characteristic matrix for C is the matrix having the elements of X as its rows. It is easy to verify that there exists at ∗ ) most one different element between ρi (Xi∗ ) and ρi+1 (Xi+1 for i = 0, 1, · · · , n − 2. Let χ(C) = {i| there exists (x0 , x1 , · · · , xn−1 ) ∈ C such that xi 0} and call χ(C) the support set of C. It was proved that |X| = |χ(C)|. Without loss of generality, we can assume |χ(C)| = n. Thus the characteristic matrix is an n × n matrix. A. Vardy and R. Koetter [4] showed the following useful result: Lemma 4: Let C be a linear code with dimension k and |χ(C)| = n, where n is the length of codewords in C. Then the spans of any two generators of C start at distinct positions and end at distinct positions. Furthermore, any minimal linear tail-biting trellis for the code C can be constructed as the product of k elementary trellises from the n characteristic generators of C. According to Lemmas 3 and 4, for a linear code C, its basis in minimal span form is important for constructing the minimal conventional trellis for C, and its characteristic generators are similarly important for constructing minimal linear tail-biting trellises for C. Definition 5: Let C1 be a linear code with length m and dimension k, and C2 a linear code with length n and dimension p. For any x = (x0 , x1 , · · · , xm−1 ) ∈ C1 , y = (y0 , y1 , · · · , yn−1 ) ∈ C2 , define x ⊗ y = (x0 y0 , x0 y1 , · · · , x0 yn−1 , · · · , xm−1 y0 , xm−1 y1 , · · · , xm−1 yn−1 ), and call x ⊗ y the tensor product of x and y. Let C1 ⊗ C2 = { xi ⊗ yi |xi ∈ C1 , yi ∈ C2 }. C1 ⊗ C2 is called the product i
code or the tensor product of C1 and C2 . By the above definition, if we view x ⊗ y as an m × n matrix with the i-th row (xi y0 , xi y1 , · · · , xi yn−1 ), then C1 ⊗C2 can be viewed as the set of matrices in which every row is an element in C2 and every column is an element in C1 . According to this structure, it is easy to see that if the distances of C1 and C2 are d1 and d2 , respectively, then the distance of C1 ⊗ C2 is d1 d2 . The tensor product of two matrices is defined as follows: for a k × m matrix G = (x1 , x2 , · · · , xk )t and a p × n matrix H = (y1 , y2 , · · · , y p )t , the tensor product G ⊗ H is a kp × mn matrix (x1 ⊗ y1 , · · · , x1 ⊗ y p , · · · , xk ⊗ y1 , · · · , xk ⊗ y p )t . Hence, if G1 and G2 are generator matrices for C1 and C2 , respectively, then G1 ⊗ G2 is a generator matrix. the equivalence of these two structures of product code was also mentioned in [8].
3.
Trellis Relations among C1 , C2 and C1 ⊗ C2
In this section, some trellis relations among C1 , C2 and C1 ⊗ C2 are discussed. Without loss of generality, we assume that all linear codes are over F2 . Let C be a linear code with length n. We always assume |χ(C)| = n. For any x ∈ C and its span (a x , b x ], (a x +i, b x + j] means ((a x + i) mod n, (b x + j) mod n by convention. If the distance of C is larger than 1. i.e., d(C) > 1, then we must have a x b x for any x ∈ C and its span (a x , b x ]. Recalling the lexicographic order ∝ in F2n , for (u0 , u1 , · · · , un−1 ), (v0 , v1 , · · · , vn−1 ) ∈ F2n , (u0 , u1 , · · · , un−1 ) ∝ (v0 , v1 , · · · , vn−1 ) iff u0 = v0 , · · · , ui−1 = vi−1 , but ui = 0 and vi = 1 for some i, 0 ≤ i ≤ n − 1, or ui = vi for all i, 0 ≤ i ≤ n − 1. Theorem 6: Let x1 , x2 , · · · , xk be a basis for C1 in minimal span form and y1 , y2 , · · · , y p a basis for C2 in minimal span form. Then: (1) xi ⊗ y j , 1 ≤ i ≤ k, 1 ≤ j ≤ p, is a basis for C1 ⊗ C2 in minimal span form. (2) If, in addition, x1 , x2 , · · · , xk is the lexicographically first basis for C1 and y1 , y2 , · · · , y p is the lexicographically first basis for C2 , Then xi ⊗ y j , 1 ≤ i ≤ k, 1 ≤ j ≤ p, is also the lexicographically first basis for C1 ⊗ C2 in minimal span form Proof. (1) Since {x1 , x2 , · · · , xk } is a basis for C1 in minimal span form, xi , 1 ≤ i ≤ k, are distinct and xi , 1 ≤ i ≤ k, are also distinct. Similarly, yi , 1 ≤ i ≤ p, are distinct and yi , 1 ≤ i ≤ p, are also distinct. Let m and n be the codeword length of C1 and C2 , respectively. For xi = (xi0 , xi1 , · · · , xi,n−1 ), by definition, xi is the smallest non-negative integer h such that xih 0 and xi the largest non-negative integer h such that xih 0. According to the definition of product code, it is easy to verify that (xi ⊗y j ) = (xi ) · n + (y j ) and (xi ⊗ y j ) = (xi ) · n + (y j ). Therefore, for any 1 ≤ i, i ≤ k, 1 ≤ j, j ≤ p, (xi ⊗ y j ) = (xi ⊗ y j ) and (xi ⊗ y j ) = (xi ⊗ y j ) iff i = i and j = j . Consequently, xi ⊗y j , 1 ≤ i ≤ k, 1 ≤ j ≤ p, is a basis for C1 ⊗C2 in minimal span form. (2) Let {x1 , x2 , · · · , xk } and {y1 , y2 , · · · , y p } be the lexicographically first bases in minimal span form for C1 and C2 , respectively. We want to prove that xi ⊗ y j , 1 ≤ i ≤ k, 1 ≤ j ≤ p, is also the lexicographically first basis for C1 ⊗C2 in minimal span form. It is enough to prove that x s ⊗ y s is lexicographically prior to w, i.e., x s ⊗ y s ∝ w, for any w ∈ C1 ⊗ C2 such that (w, w] = ((x s ⊗ y s ), (x s ⊗ y s )], where 1 ≤ s ≤ k, 1 ≤ s ≤ p, and spans are all basic spans of elements. Since xi ⊗ y j , 1 ≤ i ≤ k, 1 ≤ j ≤ p, is a basis for C1 ⊗ C2 , αi j xi ⊗ y j , (4) w= 1≤i≤k,1≤ j≤p
where αi j ∈ F2 . Since (w, w] = ((x s ⊗ y s ), (x s ⊗ y s )],
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x s ⊗y s must appear in the right of Equality (4). Furthermore, for other xi ⊗ y j appearing in the right of Equality (4), then we must have the following two cases. Case 1: xi = x s and [y j ] ⊂ [y s ], i.e., [y j ] is a proper subset of [y s ]. In this case, we get x s ⊗ y s ∝ x s ⊗ y s + x s ⊗ y j because y s ∝ y s + y j . Case 2: [xi ] ⊂ [x s ], i.e., [xi ] is a proper subset of [x s ], which means xi > x s and xi < x s . In this case, we also get x s ⊗ y s ∝ x s ⊗ y s + xi ⊗ y j because x s ∝ x s + xi . Inductively, we have x s ⊗y s ∝ w. In conclusion, xi ⊗y j , 1 ≤ i ≤ k, 1 ≤ j ≤ p, is also the lexicographically first basis for C1 ⊗ C2 in minimal span form. In the above proof, all spans mean basic spans of elements, i.e., [x] = (x, x]. Let’s recall the procedure computing the characteristic generators of a linear code. Let C be a linear code with length n and dimension k. For 0 ≤ i ≤ n − 1, let Xi∗ be the lexicographically first basis for σi (C) in minimal span form and Xi = ρi (Xi∗ ), where σ j and ρ j are cyclic shifts to the left and to the right j times, respectively. So ∪ Xi is the set of 0≤i b xi then σbxi +1 (xi ), (a xi − b xi − 1, n − 1] is an element in the lexicographically first basis for σbxi +1 (C) in minimal span form, where σbxi +1 is the cyclic shift to the left b xi + 1 times. Proof. In fact, the set {(xi , (a xi , b xi ]) | a xi ≤ b xi , 1 ≤ i ≤ n} is exactly the lexicographically first basis for C in minimal span form. For characteristic generator (xi , (a xi , b xi ]) with a xi > b xi , according to the procedure of computing the characteristic generators, we must have xi = ρ j (y), where y = n − 1 and y is an element of the lexicographically first basis for σ j (C) in minimal span form. Since [xi ] = (a xi , b xi ], j = b xi + 1. So y = σbxi +1 (xi ) and finish the proof. Before we prove the main theorem, we give an example to show the relation among characteristic matrices of linear codes C1 , C2 and C1 ⊗ C2 . Example 8: Let C1 and C2 be linear codes with the lexicographically first bases G0 and H0 in minimal span form, respectively, where
1 1 0 0 (0, 1] G0 = 0 0 1 1 (2, 3]
and
1 1 1 0 1 0 H0 = 0 0 1 0 0 0
1 1 1 0
0 1 1 1
0 0 1 1
0 1 0 1
0 0 0 1
(0, 3] (1, 6] (2, 5] , (4, 7]
and the right of matrices are the spans of corresponding rows. It is easy to get the characteristic matrices G for C1 and H for C2 , where 1 1 0 0 (0, 1] 0 0 1 1 (2, 3] G = 1 1 0 0 (1, 0] 0 0 1 1 (3, 2] and
H =
1 0 0 0 1 1 1 0
1 1 0 0 0 1 0 1
1 0 1 0 0 0 1 1
1 1 1 0 1 0 0 0
0 1 1 1 0 0 0 1
0 0 1 1 1 0 1 0
0 1 0 1 1 1 0 0
0 0 0 1 1 0 0 1
(0, 3] (1, 6] (2, 5] (4, 7] (3, 0] (6, 1] (5, 2] (7, 4]
Let x1 , x2 , x3 , x4 be the 1st, · · · , 4th rows of G, and y1 , · · · , y8 the 1st, · · · , 8th rows of H. Let Zi∗ be the lexicographically basis for σi (C1 ⊗ C2 ) in minimal span form, and Zi = ρi (Zi∗ ), where 0 ≤ i ≤ 31. So G0 ⊗ H0 is the lexicographically first basis for C1 ⊗ C2 in minimal span form. We can directly verify that (x1 ⊗ y5 , (3, 0]) ∈ Z1 , (x1 ⊗ y6 , (6, 1])) ∈ Z2 , (x1 ⊗ y7 , (5, 2]) ∈ Z3 , (x1 ⊗ y8 , (3, 2]) ∈ Z5 . Similarly, (x3 ⊗y1 , (8, 3]) ∈ Z8 , (x3 ⊗y5 , (11, 8]) ∈ Z9 , (x3 ⊗y6 , (14, 9]) ∈ Z10 . Theorem 9: Let C1 be a linear code with length m, dimension k and distance larger than 1, and C2 a linear code with length n, dimension p and distance larger than 1. Let {(xi , [xi ])|1 ≤ i ≤ m} and {(y j , [yi ])|1 ≤ j ≤ n} be the sets of the characteristic generators for C1 and C2 , respectively. Then {(xi ⊗ y j , [xi ⊗ y j ])|1 ≤ i ≤ m, 1 ≤ j ≤ n} is the set of characteristic generators of C1 ⊗ C2 , where, for [xi ] = (a, b] and [y j ] = (c, d], [xi ⊗ y j ] is computed as follows: (1) if a < b and c < d, then [xi ⊗ y j ] = (an + c, bn + d]; (2) if a < b and c > d, then [xi ⊗ y j ] = (an + c, an + d]; (3) if a > b and c < d, then [xi ⊗ y j ] = (an + c, bn + d]; (4) if a > b and c > d, then [xi ⊗ y j ] = (an + c, an + d]; Proof. For 0 ≤ i ≤ m − 1, 0 ≤ j ≤ n − 1, 0 ≤ h ≤ mn − 1, let Xi∗ , Y ∗j and Zh∗ be the lexicographically first bases for σi (C1 ), σ j (C2 ) and σh (C1 ⊗ C2 ) in minimal span form, respectively, and let Xi = ρi (Xi∗ ), Y j = ρ j (Y ∗j ) and Zh = ρh (Zh∗ ). Clearly, X0 , Y0 and Z0 are the lexicographically first bases for C1 , C2 and C1 ⊗ C2 in minimal span form, respectively. (1) By Lemma 7, If a < b and c < d, then xi ∈ X0 and y j ∈ Y0 . From Theorem 6, Z0 = X0 ⊗ Y0 . So xi ⊗ y j ∈ Z0 and its [xi ⊗ y j ] = (an + c, bn + d]. Thus (1) holds. (2) For a < b, and c > d, we try to determine a nonnegative integer h such that xi ⊗ y j ∈ Zh . Since a < b,
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(σa (xi ), (0, b − a]) ∈ Xa∗ . Since c > d, (σd+1 (y j ), (c − d − ∗ and (y j , (c, d]) ∈ Yd+1 by Lemma 7. Obvi1, n − 1]) ∈ Yd+1 ∗ ously, Xa ⊗ Y0 is the lexicographically basis for σa (C1 ) ⊗ C2 in minimal span form and σa (C1 ) ⊗ C2 = σan (C1 ⊗ C2 ). If we continually make cyclic shift on Xa∗ ⊗ Y0 to the left d + 1 ∗ such times, then there must exist an element u ∈ Zan+d+1 that [u] = (c − d − 1, mn − 1]. It is not difficult to verify that xi ⊗ y j = ρan+d+1 (u). Consequently, xi ⊗ y j ∈ Zan+d+1 and [xi ⊗ y j ] = ρan+d+1 ([u]) = (an + c, an + d]. So (2) is gotten. (3) If a > b, then (σa (xi ), (0, b−a]) ∈ Xa∗ . By Lemma 7, ∗ Zan = Xa∗ ⊗Y0 . Thus, σa (xi )⊗y j ∈ Xa∗ ⊗Y0 and [σa (xi )⊗y j ] = (c, (b − a)n + d]. Since σa (xi ) ⊗ y j = σan (xi ⊗ y j ), [xi ⊗ y j ] = ρan ([σa (xi ) ⊗ y j ]) = (an + c, bn + d]. (4) It can be gotten by similar discussion as (2) and (3). Let’s investigate example 8 again (keeping all notations in example 8). We have found the characteristic matrices G and H for C1 and C2 , respectively. By the above theorem, it is easy to get the characteristic matrix for C1 ⊗ C2 . For instance, we compute the spans of some characteristic generators of C1 ⊗ C2 , since [x1 ] = (0, 1] and [y5 ] = (3, 0], [x1 ⊗ y5 ] = (3, 0]; since [x3 ] = (1, 0] and [y1 ] = (0, 3], [x3 ⊗ y1 ] = (8, 3]; since [x3 ] = (1, 0] and [y6 ] = (6, 1], [x3 ⊗ y6 ] = (14, 9]. Now, we simply discuss the efficiency of Theorems 6 and 9. Given a basis {x1 , x2 , · · · , xk } for a linear code C1 with length m and a basis {y1 , y2 , · · · , y p } for a linear code C2 with length n, how to find the lexicographically first basis for C1 ⊗ C2 in minimal span form and its characteristic generators efficiently? If we begin from the basis {xi ⊗y j }1≤i≤k,1≤ j≤p to compute the lexicographically first basis for C1 ⊗ C2 in minimal span form and its characteristic generators by the greedy algorithm, whose time complexity is O(m2 n2 ). However, if we first compute two lexicographically first bases for C1 and C2 in minimal form from {x1 , x2 , · · · , xk } and {y1 , y2 , · · · , y p } by the greedy algorithm, respectively, then we get the lexicographically first basis for C1 ⊗ C2 in minimal span form and its characteristic generators by Theorems 6 and 9, whose time complexity is O(m2 + n2 ). Clearly, the second method is not only more efficient than the first one, but also provides the lexicographically first bases for C1 and C2 in minimal span form and their characteristic generators. Furthermore, it is more convenient to deal with shorter codes. Since the lexicographically first basis for a linear code C in minimal span form and its characteristic generators play a key role in constructing the minimal conventional trellis and minimal tail-biting trellises for C, Theorems 6 and 9 are meaningful. Remark 10: In the above theorem, we only consider linear codes with distance larger than 1, so a x b x for any codeword x and its span (a x , b x ]. For a linear code C with
the minimal distance d(C) = 1, then C = C ⊕ F2t , where d(C ) > 1 and F2t = {(x0 , x1 , · · · , xt−1 )|xi ∈ F2 , 0 ≤ i < t}. So, it is easy to get similar results of the above theorem for linear codes with distance 1. 4.
Conclusion
It is well known that the minimal conventional trellis of a linear code C can be constructed from its basis in minimal span form, and every minimal linear tail-biting trellis of C can also be constructed from its characteristic generators. In this paper, we prove that the tensor product of two lexicographically first bases for linear codes C1 and C2 in minimal span form is exactly the lexicographically first basis for C1 ⊗C2 in minimal span form, and that the tensor products of characteristic generators of C1 and C2 are exactly the characteristic generators of C1 ⊗ C2 . Formulas on basic spans of characteristic generators of C1 ⊗ C2 are given by basic spans of characteristic generators of C1 and C2 . Acknowledgement The authors are in debt to the editor and anonymous referees, whose comments greatly improve the appearance and quality of this paper. References [1] A.R. Calderbank, G.D. Forney, Jr., and A. Vardy, “Minimal tailbiting trellises: The Golay code and more,” IEEE Trans. Inf. Theory, vol.45, no.5, pp.1435–1455, July 1999. [2] H. Kan and H. Shen, “A relation between the characteristic generators of a linear code and its dual,” to appear in IEEE Trans. Inf. Theory. [3] R. Koetter and A. Vardy, “On the theory of linear trellises,” in Information, Coding and Mathematics, ed. M. Blaum, pp.323–354, Kluwer, Boston, May 2002. [4] R. Koetter and A. Vardy, “The structure of tail-biting trellises: Minimality and basic principals,” IEEE Trans. Inf. Theory, vol.49, no.9, pp.2081–2105, Sept. 2003. [5] F.R. Kschischang, “The trellis structure of maximal fixed-cost codes,” IEEE Trans. Inf. Theory, vol.42, no.6, pp.1828–1838, Nov. 1996. [6] F.R. Kschischang and V. Sorokine, “On the trellis structure of block codes,” IEEE Trans. Inf. Theory, vol.41, no.6, pp.1924–1937, Nov. 1995. [7] F.J. MacWilliams and N.J.A. Sloane, The Theory of ErrorCorrecting Codes, North-Holland, New York, 1978. [8] H. Miyagawa, Y. Iwadare, and H. Imai, Coding Theory, Shokodo, 2001. [9] A. Vardy, “Trellis structure of codes,” in Handbook of Coding Theory, ed. V.S. Pless and W.C. Huffman, pp.1989–2118, Elsevier, Amsterdam, 1998. [10] V.K. Wei and K. Yang, “On the generalized Hamming weights of product codes,” IEEE Trans. Inf. Theory, vol.39, no.5, pp.1709– 1713, Sept. 1993.
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Haibin Kan received his Ph.D. degree in 1999 from the Institute of Mathematics, Fudan University, Shanghai, China. After graduation, he became a faculty of the Department of Computer Science & Engineering of the same university, and was promoted an associate Professor in 2001. He joined JAIST as an assistant Professor in June, 2002. He has published about 15 papers in international journals. His present research interests focus on coding theory and wireless communications
Hong Shen received the B.Eng. degree from Beijing University of Science and Technology, Beijing, China, the M.Eng. degree from the University of Science and Technology of China, and the Ph.Lic. and Ph.D. degrees from Abo Akademi University, Finland, all in computer science. He is currently a full Professor in the Graduate School of Information Science, Japan Advanced Institute of Science and Technology, Ishikawa. Previously he was a Professor at Griffith University, Australia. He has published over 140 technical papers on algorithms, parallel and distributed computing, interconnection networks, parallel databases and data mining, multimedia systems, and networking. He has served as an Editor of Parallel and Distributed Computing Practice, Associate Editor of the International Journal of Parallel and Distributed Systems and Networks, a member of the editorial boards of Parallel Algorithms and Applications, International Journal of Computer Mathematics and the Journal of Supercomputing, and chaired various international conferences. Dr. Shen is a recipient of the 1991 National Education Commission Science and Technology Progress Award and the 1992 Sinica Academia Natural Sciences Award.
