L q be the determined by that of C . The following example suggests - PDF document

JOURNAL OF L A T EX CLASS FILES, VOL. 1, NO. 11, NOVEMBER 2002 1 A Classi£cation of Posets admitting MacWilliams Identity Hyun Kwang Kim and Dong Yeol Oh Remark : If P is an antichain, then P -metric is equal to Abstract — In this paper all poset structures are classi£ed which admit the MacWilliams identity, and the MacWilliams Hamming metric. So P -weight enumerator of a linear code C identities for poset weight enumerators corresponding to such becomes Hamming weight enumerator of C . posets are derived. We prove that being a hierarchical poset The MacWilliams identity for linear codes over F q is one is a necessary and suf£cient condition for a poset to admit of the most important identities in the coding theory, and MacWilliams identity. An explicit relation is also derived between it expresses Hamming weight enumerator of the dual code P -weight distribution of a hierarchical poset code and P -weight C ⊥ of a linear code C over F q in terms of Hamming weight distribution of the dual code. enumerator of C . Since Hamming metric is a special case of Index Terms — MacWilliams identity, poset codes, P -weight poset metrics, it is natural to attempt to obtain MacWilliams- enumerator, leveled P -weight enumerator, hierarchical poset. type identity for certain P -weight enumerators. See [3 - 5] for this direction of researches. Essentially, what enables us I. I NTRODUCTION to obtain MacWilliams identity for Hamming metric is that Hamming weight enumerator of the dual code C ⊥ is uniquely ET F q be the £nite £eld with q elements and F n L q be the determined by that of C . The following example suggests that vector space of n -tuples over F q . Coding theory may we need some modi£cation to generalize MacWilliams identity be considered as the study of F n q when F n q is endowed with for certain type of poset weight enumerators. Hamming metric. Since the late 1980’s several attempts have Example 1.1: Let P = { 1 , 2 , 3 } be a poset with order been made to generalize the classical problems of the coding relation 1 < 2 < 3 and P = { 1 , 2 , 3 } be a poset with order theory by introducing a new non-Hamming metric on F n q (cf relation 1 > 2 > 3 . Consider the following binary linear P - [8 - 10]). These attempts led Brualdi et al. [1] to introduce the codes: concept of poset codes. First, we begin by brie¤y introducing C 1 = { (0 , 0 , 0) , (0 , 0 , 1) } , C 2 = { (0 , 0 , 0) , (1 , 1 , 1) } . the basic notions of poset code such as poset-weight and poset- It is easy to check that P -weight enumerators of C 1 and C 2 distance. See [1] for details. are given by Let F n q be the vector space of n -tuples over a £nite £eld F q W C 1 , P ( x ) = 1 + x 3 = W C 2 , P ( x ) . with q elements. Let P be a partial ordered set, which will The dual codes of C 1 and C 2 are respectively given by be abbreviated as a poset in the sequel, on the underlying set C ⊥ 1 = { (0 , 0 , 0) , (1 , 0 , 0) , (0 , 1 , 0) , (1 , 1 , 0) } [ n ] = { 1 , 2 , . . . , n } of coordinate positions of vectors in F n and q with the partial order relation denoted by ≤ as usual. For u = C ⊥ 2 = { (0 , 0 , 0) , (1 , 1 , 0) , (1 , 0 , 1) , (0 , 1 , 1) } . ( u 1 , u 2 , · · · , u n ) ∈ F n q , the support supp ( u ) and P -weight The P -weight enumerators of C ⊥ 1 and C ⊥ 2 are given by 2 , P ( x ) = 1 + x 2 + 2 x 3 , w P ( u ) of u are de£ned to be 1 , P ( x ) = 1 + x + 2 x 2 , W C ⊥ W C ⊥ � while P -weight enumerators of C ⊥ 1 and C ⊥ 2 are given by � u i � = 0 } and w P ( u ) = | < supp ( u ) > | , supp ( u ) = { i 1 , P ( x ) = 1 + x 2 + 2 x 3 = W C ⊥ W C ⊥ 2 , P ( x ) . where < supp ( u ) > is the smallest ideal (recall that a subset As it is seen above, although P -weight enumerators of the I of P is an ideal if a ∈ I and b ≤ a , then b ∈ I ) containing codes C 1 and C 2 are the same, P -weight of the dual codes the support of u . It is well-known that for any u, v ∈ F n q , may be different. Fortunately,however, P -weight enumerators d P ( u, v ) := w P ( u − v ) is a metric on F n q . The metric d P is of the dual codes are the same. called P -metric on F n q . Let F n q be endowed with P -metric. Feeding back this information we de£ne, for a given poset Then a (linear) code C ⊆ F n q is called a (linear) P -code of P , the poset P as follows: length n . The P -weight enumerator of a linear P -code C is P and P have the same underlying set and de£ned by x ≤ y in P ⇔ y ≤ x in P . W C , P ( x ) = � � n x w P ( u ) = A i, P x i , The poset P is called the dual poset of P . u ∈C i =0 � De£nition 1.2: Let P be a poset on [ n ] . It is said that P � w P ( u ) = i }| . where A i, P = |{ u ∈ C admits MacWilliams identity if P -weight enumerator of the dual code C ⊥ of a linear code C over F q is uniquely determined This research was supported by the Com 2 MaC-KOSEF and POSTECH by P -weight enumerator of C . BSRI research fund. For an illustration of our de£nition, we give two classes of H.K.Kim and D.Y.Oh are with the Department of Mathematics, Pohang posets which admit MacWilliams identity. University of Science and Technology, Pohang 790-784, Korea.