Composition of Gray Isometries
11 COMPOSITION OF GRAY ISOMETRIES
SIERRA MARIE M. LAURESTA and Virgilio P. Sison
Institute of Mathematical Sciences and Physics University of the Philippines Los Baños College, Laguna 4031 E-mail: [email protected], [email protected]
Abstract
In coding theory, Gray isometries are usually defined as mappings between finite Frobenius rings, which include the ring ℤ 𝑚 of integers modulo m and the finite fields. In this paper, we derive an isometric mapping from ℤ to ℤ from the composition of the Gray isometries on ℤ and on ℤ . The image under this composition of a ℤ -linear block code of length n with homogeneous distance d is a (not necessarily linear) quaternary block code of length with Lee distance d . Introduction A (block) code of length n over a ring R is a non-empty set of n -tuples over R called codewords . It is said to be linear of rate- k/n if it is a (not necessarily free) submodule, and is completely determined by a 𝑘 × 𝑛 matrix G over R . Codes over rings gained more attention when Hammons, Kumar, Calderbank, Sloane and Solé [1] discovered in 1994 that certain very good but peculiar nonlinear codes over the binary field 𝔽 can be viewed as images of linear codes over the integer ring ℤ under the so-called Gray map 𝜙 from ℤ onto 𝔽 defined by and The map 𝜙 is an isometry or is weight-preserving, that is, the Lee weight of an element of ℤ is equal to the Hamming weight of its image under 𝜙 . The Hamming weight of a binary vector is the number of nonzero components in the vector. Carlet [2] introduced a generalization of 𝜙 to the ring ℤ 𝑘 of integers modulo 𝑘 . Let k be a positive integer , u an element of ℤ 𝑘 and 𝑢 = ∑ 2 𝑖−1 𝑢 𝑖𝑘𝑖=1 its 2-adic expansion, where 𝑢 𝑖 ∈ 𝔽 . The image of u by the generalized Gray map is the boolean function on 𝔽 given by (𝑦 , 𝑦 , … , 𝑦 𝑘−1 ) ⟶ 𝑢 𝑘 + ∑ 𝑢 𝑖 𝑦 𝑖𝑘−11=1 . We identify this boolean function with a binary word of length 𝑘−1 by simply listing its values. Thus the generalized Gray map is seen as a nonsurjective mapping from ℤ 𝑘 to 𝔽 𝑘−1 , and its image is the Reed-Muller code of order 1,
RM(1, k-1).
The generalized Gray map is naturally extended to the n- tuples. When k = 2, RM(1,1) is the set of boolean functions on 𝔽 , and we obtain the usual Gray map 𝜙 from ℤ to 𝔽 . When k = 3 , the generalized Gray map (which we denote by 𝜓 ) takes ℤ onto RM(1,2) which is the set of boolean functions on 𝔽 that give all the binary words in 𝔽 with even Hamming weight. Methodology
We extend the usual Gray isometry 𝜙 as a bijective mapping from ℤ onto 𝔽 . Table 1 shows the binary image of an element of ℤ under 𝜙 . Clearly the Lee weight 𝑤 𝐿 of an element of ℤ is equal to the Hamming weight 𝑤 𝐻 of its binary image. (𝑥, 𝑦) ∈ ℤ 𝑤 𝐿 (𝑥, 𝑦) 𝜙(𝑥, 𝑦) ∈ 𝔽 𝑤 𝐻 (𝜙(𝑥, 𝑦)) (0,0) 0 (0,0,0,0) 0 (0,1) 1 (0,0,0,1) 1 (0,2) 2 (0,0,1,1) 2 (0,3) 1 (0,0,1,0) 1 (1,0) 1 (0,1,0,0) 1 (1,1) 2 (0,1,0,1) 2 (1,2) 3 (0,1,1,1) 3 (1,3) 2 (0,1,1,0) 2 (2,0) 2 (1,1,0,0) 2 (2,1) 3 (1,1,0,1) 3 (2,2) 4 (1,1,1,1) 4 (2,3) 3 (1,1,1,0) 3 (3,0) 1 (1,0,0,0) 1 (3,1) 2 (1,0,0,1) 2 (3,2) 3 (1,0,1,1) 3 (3,3) 2 (1,0,1,0) 2 Table 1: The isometric Gray map 𝝓 on ℤ We restrict 𝜙 −1 as a mapping from RM(1,2) to ℤ as follows: 𝑥 ∈ 𝑅𝑀(1,2) 𝜙 −1 (𝑥) (0,0,0,0) (0,0) (0,1,0,1) (1,1) (0,0,1,1) (0,2) (0,1,1,0) (1,3) (1,1,1,1) (2,2) (1,0,1,0) (3,3) (1,1,0,0) (2,0) (1,0,0,1) (3,1) Table 2: The map 𝝓 −𝟏 on 𝑹𝑴(𝟏, 𝟐)
For ℤ we apply the following homogeneous weight [3] and extend it coordinatewisely. 𝑤 ℎ𝑜𝑚 (𝑥) = {0 𝑖𝑓 𝑥 = 04 𝑖𝑓 𝑥 = 42 𝑒𝑙𝑠𝑒 Table 3 shows the image of an element of ℤ in RM(1,2) under the generalized Gray map 𝜓. If 𝑢 ∈ ℤ has the 2-adic expansion 𝑢 = 𝑢 + 2𝑢 + 4𝑢 , then 𝜓(𝑢) = 𝜓(𝑢 + 2𝑢 + 4𝑢 ) =(𝑢 , 𝑢 + 𝑢 , 𝑢 + 𝑢 , 𝑢 + 𝑢 + 𝑢 ) . The mapping 𝜓 is weight preserving such that the homogeneous weight of an element of ℤ is equal to the Hamming weight of its image in RM(1,2) . 𝑢 ∈ ℤ 𝑤 ℎ𝑜𝑚 (𝑢) 𝜓(𝑢) ∈ 𝑅𝑀(1,2) 𝑤 𝐻 (𝜓(𝑢))
0 0 (0,0,0,0) 0 1 2 (0,1,0,1) 2 2 2 (0,0,1,1) 2 3 2 (0,1,1,0) 2 4 4 (1,1,1,1) 4 5 2 (1,0,1,0) 2 6 2 (1,1,0,0) 2 7 2 (1,0,0,1) 2
Table 3: The isometric Gray map 𝝍 on ℤ Results and Discussion
We take the composition 𝜙 −1 𝜓: ℤ ⟶ ℤ . Table 4 shows the quaternary image of an element of ℤ under 𝜙 −1 𝜓. If 𝑢 = 𝑢 + 2𝑢 + 4𝑢 ∈ ℤ , then 𝜙 −1 𝜓(𝑢) = 𝜙 −1 𝜓(𝑢 + 2𝑢 + 4𝑢 ) = (𝑢 +2𝑢 , 𝑢 + 2𝑢 + 2𝑢 ) . 𝑢 ∈ ℤ 𝑤 ℎ𝑜𝑚 (𝑢) 𝜙 −1 𝜓(𝑢) 𝑤 𝐿 (𝜙 −1 𝜓(𝑢))
0 0 (0,0) 0 1 2 (1,1) 2 2 2 (0,2) 2 3 2 (1,3) 2 4 4 (2,2) 4 5 2 (3,3) 2 6 2 (2,0) 2 7 2 (3,1) 2
Table 4: The isometric map 𝝓 −𝟏 𝝍: ℤ → ℤ The mapping 𝜙 −1 𝜓 is weight preserving such that the homogeneous weight of an element of ℤ is equal to the Lee weight of its image in ℤ . It is extended naturally to the n -tuples. Let C be a linear block code of length n over ℤ with minimum homogeneous distance d. The image of C under 𝜙 −1 𝜓 is the set 𝜙 −1 𝜓(𝐶) = {𝑤 ∈ ℤ | 𝑤 = 𝜙 −1 𝜓(𝑐) for 𝑐 ∈ 𝐶}. Proposition . The set 𝜙 −1 𝜓(𝐶) has the following properties: i. 𝜙 −1 𝜓(𝐶) is a (not necessarily linear) block code of length over ℤ . ii. The Lee distance of 𝜙 −1 𝜓(𝐶) is equal to d. iii. Every codeword of 𝜙 −1 𝜓(𝐶) has even Lee weight. To illustrate non-linearity, consider the rate-2/3 linear block code over ℤ generated by the matrix 𝐺 = (1 2 70 2 4)
This code has 32 codewords, minimum Hamming distance 1, and minimum homogeneous distance 4. The codewords (6,6,6) and (7,6,1) generated by the information words (6,1) and (7,4), respectively, have quaternary images (2,0,2,0,2,0) and (3,1,2,0,1,1) whose superimposition is not in the code. This example also shows that 𝜙 −1 𝜓 is not an additive homomorphism. Conclusion and Recommedation