Ternary Codes From Modified Jacket Matrices - IEEE Xplore

12

JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 13, NO. 1, FEBRUARY 2011

Ternary Codes From Modified Jacket Matrices Xueqin Jiang, Moon Ho Lee, Ying Guo, Yier Yan and Sarker Md. Abdul Latif ∗ ˜ ∗ of Abstract: In this paper, we construct two families Cm and C m m m m−1 m m+1 m−1 ) and (2 , 3 ,2 ) codes, for m = ternary (2 , 3 , 2 1, 2, 3, · · ·, derived from the corresponding families of modified ternary Jacket matrices. These codes are close to the Plotkin bound and have a very easy decoding procedure.

Index Terms: Algebraic integers, cyclotomic fields, Jacket codes, Kronecker products of matrices, modified ternary, modified ternary Jacket matrices.

I. INTRODUCTION Many discrete signal transforms are based on the use of transform matrices with entries on the complex circle, such as the family of discrete generalized transforms (DGT) for signals of length n = 2m [1, 10.2]. This family includes the Walsh-Hadamard transform (WHT) and the 2m -point discrete Fourier transform (DFT). Interpretation of the Cooley-Turkey fast Fourier transform (FFT) in terms characters of abelian groups [2], [3] means that the DFT is itself a generalized transform which includes the WHT. Both of the WHT and DFT are suboptimal discrete orthogonal transforms, but each has wide application. The above transform matrices belong to the more general matrices, Jacket matrices, which are motivated by the center weighted Hadamard matrices [4]. In a general definition, any square matrix [J] = [js,t ]n×n is called a Jacket matrix if its inverse matrix is obtained simply by an element-wise inverse [5]–[7], namely, [J]

−1

T  1 1 = c js,t n×n

for 1 ≤ s, t ≤ n where T denotes the transpose of the matrix, c is the normalized constant. The cyclic function on Jacket matrices will be help in signal processing [8], sequence design, cryptography [9], and quantum information [7]. ˜ m } of In this paper, we consider two families {Mm } and {M modified ternary Jacket matrices and construct the correspond∗ ∗ ∗ } and {C˜m } of nonlinear ternary codes Cm ing families {Cm Manuscript received September 26, 2009; approved for publication by Emanuele Viterbo, Division I Editor, August 10, 2010. This work was supported by World Class University R32-2008-000-20014-0 NRF, Korea. X. Jiang is with the School of Information Science and Technology of Donghua University, China, email: [email protected]. M. Lee is with the Division of Electronic and Information Engineering of Chonbuk National University, Korea, email: [email protected]. Y. Guo is with the Central South University, China, email: yingguo@mail. csu.edu.cn. Y. Yan is with School of Mechanical and Electrical Engineering of Guangzhou University, China, email: [email protected]. S. M. A. Latif is with the Electronics Engineering from Chonbuk National University, Korea, email: [email protected].

∗ ˜ m , respectively. and C˜m , derived from the matrices Mm and M The parameters of these codes are described as follows. ∗ ∗ and C˜m have parameters Theorem 1: The ternary codes Cm

(2m , 3m , 2m−1 ) and (2m , 3m+1 , 2m−1 )  m−1  respectively, and correct t ≤ 2 2 −1 errors. The rest of this paper is organized as follows. In Section II, we will introduce a family of modified Jacket matrix. In Section III, ∗ ∗ } and {C˜m } of nonlinear pwe introduce two families {Cm ary codes. Section IV introduces the encoding procedure and Section V introduces the decoding algorithm. In Section VI, we give an example of a ternary Jacket code. Finally, Section V concludes the paper. II. MODIFIED TERNARY JACKET MATRICES Let ω = e2πi/3 be a primitive cubic root of unity, and Q(ω) the cyclotomic field obtained from the field of rational numbers Q by enjoining of ω. The field Q(ω) is a quadratic extension of Q, and the minimal polynomial of ω over Q is f (x) = x2 + x + 1.

(1)

The elements 1, ω form a basis of Q(i) over Q, so any α ∈ Q(ω) can be uniquely written as a linear combination α = a + bω of the basis elements 1 and ω with coefficients a, b ∈ Q. Let Z be the ring of rational integers. In this paper, we work in the ring Z[ω] of algebraic integers of Q(i). The elements of Z[ω] are algebraic integers of the form α = a + bω, where a, b ∈ Z. The ring Z[ω] contains the multiplicative cyclic group G = {1, ω, ω 2} of order 3. Consider the Jacket matrix ⎞ ⎛ 1 1 1 (2) J = ⎝ 1 ω ω2 ⎠ 1 ω2 ω and then use the elements of J to compose a new matrix M1 as follows ⎛ ⎞ 1 1 M1 = ⎝ 1 ω ⎠ . (3) ω 1 The Jacket matrix J in (2) is a square and symmetric matrix and JJ ∗ = nI where J ∗ is the Hermite transpose of J. We extend this property to a new nonsquare and nonsymmetric matrix M1 . M1 is similar to J in many aspects which will be shown by (10) and property 2.1. However, M1 is not a Jacket matrix. It is a matrix obtained from the Jacket matrix in (2). ˜ m , for m = We define a modified ternary matrices Mm and M 2, 3, · · ·, respectively by the relations ⎞ ⎛ Mm−1 Mm−1 Mm = M1 ⊗ Mm−1 = ⎝ Mm−1 ωMm−1 ⎠ (4) ωMm−1 Mm−1

c 2011 KICS 1229-2370/11/$10.00 

JIANG et al.: TERNARY CODES FROM MODIFIED JACKET MATRICES

and

⎞ Mm = ⎝ ωMm ⎠ . ω 2 Mm

13

⎛

˜m M

(5)

˜ m are 3m × 2m and 3m+1 × 2m matrices, Clearly, Mm and M respectively, with entries from the cyclic group G =< ω >. If

 1 1 ω2 ∗ M1 = (6) 1 ω2 1 is the Hermite transpose of M1 , we set ⎛ 2 1 + ω2 ∗ ⎝ 1+ω 2 D1 = M 1 M 1 = 1 + ω ω + ω2

⎞ 1 + ω2 ω + ω2 ⎠ . 2

(7)

Taking into account that 1 + ω + ω 2 = 0, we can rewrite D1 in the form ⎛ ⎞ 2 −ω −ω 2 −1 ⎠ . D1 = ⎝ ω 2 (8) 2 −1 2 −ω It is easy to see that D1 is a self-conjugate complex matrix, that is D1∗ = D1 . Finally, we defined complex self-conjugate matrices Dm , for m = 2, 3, · · ·, recursively by Dm = D1 ⊗ Dm−1 ⎛ 2Dm−1 = ⎝ −ω 2 Dm−1 −ω 2 Dm−1

−ωDm−1 2Dm−1 −Dm−1

⎞ −ωDm−1 −Dm−1 ⎠ . (9) 2Dm−1

Since M1 M1∗ = D1 , it follows that ∗ Mm Mm = Dm .

(10)

The last relation show that the matrices Mm , for m = 1, 2, · · ·, do not fall into the class of Jacket matrices. On the other hand, it is easy to see, using induction on m, that the following holds. Proposition 1: Let Dm = (θkl ). For any m ≥ 1 we have ∗ Dm = Dm . The entries θkl = akl + bkl ω of Dm are algebraic integers form the ring Z[ω]. The diagonal elements θk,k of Dm all are equal to 2m , and the first coefficients akl of the entries θkl lying outside of the diagonal do not exceed 2m−1 . The above Proposition and relations (10) show that the modified matrices Mm in many aspects are very similar to the corresponding Jacket matrices. This fact provides a very easy and ∗ ∗ and C˜m , sufficiently fast decoding algorithm for the codes Cm described below in Section IV. III. MODIFIED TERNARY JACKET CODES ˜ m be modified ternary Jacket matrices defined Let Mm and M ∗ is defined as the as above. A modified ternary Jacket code Cm ∗ of Mm . The set of all columns of the Hermite transpose Mm ∗ columns of Mm can be indexed by the integers from 0 to ∗ is the 3m − 1. Similarly, a modified ternary Jacket code C˜m ∗ ˜ set of all columns of the Hermitian transpose Mm of the ma∗ ˜m ˜ m . The columns of M can be indexed by the integers trix M m+1 ∗ ∗ − 1. It is clear that Cm and C˜m are nonlinear from 0 to 3

(2m , 3m , d) and (2m , 3m+1 , d) codes, respectively. Let us find ∗ ∗ the minimum Hamming distances d of the codes Cm and C˜m . Theorem 2: The minimal Hamming distances d of the codes ∗ ∗ Cm and C˜m is equal to 2m−1 . Proof: It is clear that the minimal Hamming distance of ∗ ∗ Cm and C˜m are equal to the minimum distance between distinct ˜ m , respectively. rows of the matrices Mm and M To prove that this minimal distance is 2m−1 , we use induction in m. The statement is clearly true for m = 1. Let now m ≥ 2. Suppose that the minimum Hamming distance between distinct rows of Mm−1 is equal to 2m−2 and prove that the minimum Hamming distance between distinct rows of Mm equals 2m−1 . Write ⎞ ⎛ Mm−1 Mm−1 Mm = ⎝ Mm−1 ωMm−1 ⎠ ωMm−1 Mm−1 and consider the following submatrices M (1,1) = (Mm−1 Mm−1 ), M (1,ω) = (Mm−1 ωMm−1 ) and

M (ω,1) = (ωMm−1 , Mm−1 )

of the matrix Mm . By the induction hypothesis, the minimum distance between distinct rows of Mm−1 is equal to 2m−2 . Consider the matrix Mm−1 as an ordered set of its rows. If (ak , ak ) and (al , al ) are two distinct rows of Mm−1 (1, 1), then the Hamming distance between (ak , ak ) and (al , al ) equals d(ak , al ) + d(ak , al ) where d(ak , al ) is the Hamming distance between two distinct rows ak and al of the matrix Mm−1 . This shows that the minimum Hamming distance between distinct (1,1) rows of Mm−1 is equal to 2m−1 . Similarly, the minimum Hamming distance between distinct rows of each matrix M (1,ω) and M (ω,1) is equal to 2m−1 . Now we proceed as follows. (i) First we show that the minimum Hamming distance be(1,1) (1,ω) tween any two rows of the submatrices Mm−1 and Mm−1 is m−1 . Let (ak , ak ) and (al , ωal ) be two arbitrary rows at least 2 (1,1) (1,ω) of Mm−1 and Mm−1 , respectively. If l = k, then the Hamming distance between (ak , ak ) and (al , ωal ) is at least 2m−1 . Let now l = k. The Hamming distance between (ak , ak ) and (al , ωal ) equals d(ak , al ) + d(ak , ωal ). Using the triangle inequality d(al , ωal ) ≤ d(al , ak ) + d(ak , ωal ) we find that the Hamming distance between (ak , ak ) and (al , ωal ) is at least d(al , ωal ) = 2m−1 . Thus, the minimum (1,1) Hamming distance between any two rows of Mm−1 , respecm−1 tively, is at least 2 . Similarly, the minimum Hamming dis(1,1) (ω,1) tance between the rows of Mm−1 and Mm−1 , respectively, is at least 2m − 1. (ii) Now we prove that the minimum Hamming distance be(1,ω) (ω,1) tween any two rows of the submatrices Mm−1 and Mm−1 again m−1 . Consider two arbitrary elements (ak , ωak ) ∈ is at least 2 (1,ω) (ω,1) Mm−1 and (ωal , al ) ∈ Mm−1 . If l = k then the Hamming distance between (ak , ωak ) and (ωal , al ) is at least 2m−1 . If l = k, we have d(ωal , al ) ≤ d(ωal , ak ) + d(ak , ωal )

14

JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 13, NO. 1, FEBRUARY 2011

and since d(ωal , ak ) = d(ak , ωal ) and d(ak , ωal ) = d(ωak , al ) then d(ωal , al ) ≤ d(ak , ωal ) + d(ωak , al ). Thus, the Hamming distance between (ak , ωak ) and (ωal , al ) is at least d(ωal , al ) = 2m−1 . This completes the proof.  ∗ ∗ and C˜m have Corollary 1: The nonlinear ternary codes Cm parameters (2m , 3m , 2m−1 ) and (2m , 3m+1 , 2m−1 ), respectively. IV. ENCODING ALGORITHM If we would like to transmit symbols 0, 1, and 2, we can transform these symbols to 1, ω, and ω 2 using the following one-toone map bk = ω ik . (11) In fact, the multiplicative cyclic group G = {1, ω, ω 2} of order 3 is isomorphic to the set {0, 1, 2} with modulo 3. Each ∗ carries m symbols of informamodified ternary Jacket code Cm tion. Given the input information sequence (i0 , i1 , · · ·, i(m−1) ) where ik ∈ {0, 1, 2}, for k = 0, 1, · · ·, m − 1. We obtain (b0 , b1 , · · ·, b(m−1) ) from the one-to-one map (11). The encod∗ includes two steps: ing procedure of Cm Given b0 , calculate aτ0 with the fomular (1, b20 )τ , b0 ∈ {1, ω} τ a0 = (12) (ω 2 , 1)τ , b0 = ω 2 and calculate aτ(k) from aτ(k−1) recursively with the formula

 aτ(k+1)

=

ak , b 2 ak τ  2 (k+1) ω ak , ak ,

τ

, b(k+1) ∈ {1, ω} b(k+1) = ω 2

(13)

where τ denotes the transpose of a vector. Actually, given the information sequence (i0 , i1 , · · ·, im−1 ), the result code-vector ∗ ∗ is the ith column of Mm where i = i0 + i1 3 + aτ(m−1) ∈ Cm m−1 . · · · + i(m−1) 3 Let us illustrate the encoding process by using an example. Assume m = 2 and the information sequence is (i0 , i1 ) = (0, 2). From (3) and (4), we have the following ternary matrix ⎛ ⎞ 1 1 1 1 ⎜ 1 ω 1 ω ⎟ ⎜ ⎟ ⎜ ω 1 ω 1 ⎟ ⎜ ⎟ ⎜ 1 1 ω ω ⎟ ⎜ ⎟ ω ω ω2 ⎟ M2 = ⎜ ⎜ 1 ⎟ ⎜ ω 1 ω2 ω ⎟ ⎜ ⎟ ⎜ ω ω 1 1 ⎟ ⎜ ⎟ ⎝ ω ω2 1 ω ⎠ 1 ω2 ω ω and its Hermitian transpose ⎛ 1 1 ω2 1 ⎜ 1 ω2 1 1 M2∗ = ⎜ ⎝ 1 1 ω2 ω2 1 ω2 1 ω2

1 ω2 ω2 ω

ω2 1 ω ω2

ω2 ω2 1 1

ω2 ω 1 ω2

⎞ ω ω2 ⎟ ⎟. ω2 ⎠ 1

From (11), we get (b0 , b1 ) = (1, ω 2 ). With (12) and b0 = 1, we get a0 = (1, 1). Then, with (13) and b1 = ω 2 , a0 = (1, 1), we get the finally code word a1 = (ω 2 , ω 2 , 1, 1). Actually, from i = i0 + i1 3 = 6, we can also see that the final code word is the 6th column of M2∗ , which confirms our encoding algorithm. ∗ Each modified ternary Jacket code C˜m carries (m + 1) symbols of information. Given the input information sequence (b0 , b1 , · · ·, b(m−1) , bm ) where bi ∈ {1, ω, ω 2}, for i = ∗ 0, 1, · · ·, m. The encoding procedure of C˜m includes two steps. τ First, code-vector a(m−1) can be obtained recursively based on ∗ can be ob(12) and (13). Second, the code-vector aτm ∈ C˜m tained by (14) aτm = ¯bm aτ(m−1) where ¯bm is the complex conjugate of bm . Then, the result code∗ ∗ ˜m is the ith column of M where i = i0 + i1 3 + vector aτm ∈ C˜m m−1 m · · · + i(m−1) 3 + im 3 . V. DECODING ALGORITHM ∗ ∗ The codes Cm and C˜m admit an effective decoding proce∗ ∗ and C˜m are very similar dure. Decoding algorithms for Cm and we restrict ourselves by description of the decoding algo∗ . Let Mm = (ai,j ), 1 ≤ i ≤ 3m , rithm for the code Cm m 1 ≤ j ≤ 2 be a modified 3m × 2m Jacket matrix, a transmitted ∗ code-vector a ¯τi = (¯ ai,1 , · · ·, a ¯i,2m )τ ∈ Cm , and a received vecτ ci,1 , · · ·, c¯i,2m ) that differs from a ¯τi in t positions. We tor c¯i = (¯ assume that the noisy channel can transform each symbol a ¯i,j from the alphabet G = {1, ω, ω 2 } to some another symbol c¯i,j from G with the same small probability p∗ and leaves a ¯i,j fixed with probability 1 − p∗ . To restore the transmitted vector a ¯τi from received vector c¯τi , there are three steps in the decoding process. First, we multiply the matrix Mm by c¯τi and then resulting vectors sτ = Mm c¯τi . Since the entries of Mm and the components of a ¯τi are elements 2 of the cyclic group G = {1, ω, ω }, the resulting vector sτ = (s1 , · · ·, s3m )τ is a vector of size 3m , whose components sk , for 1 ≤ k ≤ 3m , are elements of the ring Z[ω]. This means that each component sk is a linear combination (0)

(1)

sk = sk + sk ω (0)

(1)

of elements 1 and ω with coefficients sk ,sk ∈ Z. Secondly, to correct possible errors we example the components of the syndrome sτ = (s1 , · · ·, s3m )τ . If the number of distorted symbols in the received vector is    m−1  d−1 2 −1 t≤ = 2 2 then among the components sk , 1 ≤ k ≤ 3m , of the vector s, we (0) choose the unique one si whose first coefficient si is strictly greater than the first coefficient of any other component sk of s. We notice that if no error occurs then si ∈ Z and si has the maximal possible value 3m . Thirdly, we decode c¯τi as the transmitted vector a ¯τi = τ ¯i,2m ) . In other words, the received vector c¯i is de(¯ ai,1 , · · ·, a coded as the complex conjugate a ¯i of the ith row of the modified

JIANG et al.: TERNARY CODES FROM MODIFIED JACKET MATRICES

15

ternary Jacket matrix Mm . Then, the input information sequence is (i0 , i1 , · · ·, i(m−1) ) where i = i0 + i1 3 + · · · + i(m−1) 3m−1 . ∗ The decoding process is finished.  It is clear that the code Cm  d−1  m−1 corrects t ≤ 2 = 2 2 −1 errors. Similarly, the code C˜m with parameters (2m , 3m+1 , 2m−1 ) m−1 also corrects any t ≤ 2 2 −1 errors.

The ternary code C3∗ consists of the columns of 33 × 23 matrix M3∗ and has parameters (23 , 33 , 22 ). Let us show that the code C3∗ corrects single errors. Consider a code-vector aτ ∈ C3∗ , say aτ = (ω 2 , 1, ω 2 , 1, ω 2 , 1, ω 2 , 1)τ and assume that this vector is sent through a noisy channel. Let cτ = (ω 2 , 1, ω 2 , 1, ω 2 , 1, ω 2 , ω)τ

VI. AN EXAMPLE Again, we consider the matrix M2 and its Hermitian transpose M2∗ . In view of (10), we have

be the received vector which differs from aτ in the last position. To correct the error, we multiply M3 by cτ and then take into account the relation 1 + ω + ω 2 = 0.

M2 M2∗ = D2 As a result, we obtain

where D2 is shown at the top of next page. Let now ⎞ ⎛

M2 M2 M2∗ M2∗ M3 = ⎝ M2 ωM2 ⎠ , M3∗ = ∗ M2 ω 2 M2∗ ωM2 M2 and

⎛

2M2 D3∗ = ⎝ −ω 2 M2 −ω 2 M2

−ωM2 2M2 −M2

M 3 cτ = sτ ω 2 M2∗ M2∗



where s = (−1 − 3ω, −5 − 2ω, 7 + ω, 1 − 2ω, −ω, 3 + 2ω, 1 + ω, −3 − 4ω, 3 + 5ω, 1 − 2ω, −ω, 3 + 2ω, 3 + 2ω, −1, 2 + 3ω, −1, 2 + 3ω, −ω, 2, −1, 1 + ω, −3 − 4ω, 3 + 5ω, −ω, 2, −1, 2ω, −1 − 3ω, −1 + ω)

⎞ −ωM2 −M2 ⎠ 2M2

The components of the syndrome sτ are elements

so that

M3 M3∗ = D3

(0)

si = si

and then we have the following 27 × 8 matrix ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ M3 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1 1 ω 1 1 ω ω ω ω2 1 1 ω 1 1 ω ω ω ω2 ω ω ω2 ω ω ω2 ω2 ω2 1

1 ω 1 1 ω 1 ω ω2 ω 1 ω 1 1 ω 1 ω ω2 ω ω ω2 ω ω ω2 ω ω2 1 ω2

1 1 ω ω ω ω2 1 1 ω 1 1 ω ω ω ω2 1 1 ω ω ω ω2 ω2 ω2 1 ω ω ω2

1 ω 1 ω ω2 ω 1 ω 1 1 ω 1 ω ω2 ω 1 ω 1 ω ω2 ω ω2 1 ω2 ω ω2 ω

1 1 ω 1 1 ω ω ω ω2 ω ω ω2 ω ω ω2 ω2 ω2 1 1 1 ω 1 1 ω ω ω ω2

1 ω 1 1 ω 1 ω ω2 ω ω ω2 ω ω ω2 ω ω2 1 ω2 1 ω 1 1 ω 1 ω ω2 ω

1 1 ω ω ω ω2 1 1 ω ω ω ω2 ω2 ω2 1 ω ω ω2 1 1 ω ω ω ω2 1 1 ω

1 ω 1 ω ω2 ω 1 ω 1 ω ω2 ω ω2 1 ω2 ω ω2 ω 1 ω 1 ω ω2 ω 1 ω 1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(1)

+ si ω (0)

of the ring Z[ω]. Since the first coefficient s3 = 7 of the element 7 + ω in 3rd position of sτ is strictly greater than the first coefficient of any other component of sτ , we decode the received vector cτ as the vector aτ = (ω 2 , 1, ω 2 , 1, ω 2 , 1, ω 2 , 1)τ from the 3rd column of the matrix M3∗ Now we assume that aτ = (ω, ω, ω 2 , ω 2 , ω 2 , ω 2 , 1, 1)τ is a transmitted code-vector, and cτ = (ω, ω, ω, ω 2 , ω 2 , ω 2 , 1, 1)τ is the received vector. Multiplying the matrix M3 by cτ we obtain M 3 cτ = sτ where s = (−1, 1 + ω, −2 − 2ω, −2 + ω, −3 − ω, −ω, 1 − 2ω, 3 + 2ω, −ω, 1 + 4ω, 2ω, −3 − ω, −ω, −1, 2, −3 + 2ω, 2ω, −4 − 3ω, −2 − 5ω, −ω, 3 − ω, −3 − ω, −1 − 3ω, −1, 6 − ω, 2 + 3ω, 5 + 3ω) (0)

Again, the first coefficient s25 = 6 of the element 6 − ω in 25th position of sτ is strictly greater than the first coefficient of any other component of sτ , so we decode the received vector cτ as the vector aτ = (ω, ω, ω 2 , ω 2 , ω 2 , ω 2 , 1, 1)τ from the 25th column of the matrix M3∗ . In other words, we decode cτ as the complex conjugate of the 25th column of the matrix M3∗ . Similarly, it is easy to see that the ternary code c˜∗3 with parameters (23 , 34 , 22 ) also corrects any single errors.

16

JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 13, NO. 1, FEBRUARY 2011

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ D2 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

22 −2ω 2 −2ω 2 −2ω 2 ω ω −2ω 2 ω ω

−2ω 22 −2 1 −2ω 2 ω2 1 −2ω 2 ω2

−2ω −2 22 1 ω2 −2ω 2 1 ω2 −2ω 2

−2ω 1 1 22 −2ω 2 −2ω 2 −2 ω2 ω2

VII. CONCLUSION In this paper, we consider a family {Mm }, m = 1, 2, · · ·, of modified ternary Jacket matrix of order 3m . We construct two ∗ ∗ } and {C˜m } of nonlinear 3-ary codes derived from families {Cm Kronecker powers Mm = M1⊗m of the modified ternary Jacket Matrix. These codes are close to the Plotkin bound and have nice parameters and very easy encoding and decoding procedures. REFERENCES [1] D. F. Elliot and K. R. Rao, “Fast transforms: Algorithms, analyses,” Applications, Academic Press, New York, 1982. [2] D. K. Maslen and D. N. Rockmore, “Generalized FFTs-a survey of some recent results,” DIMACS Ser. Discr. Math. Theoret. Comp. Sci., vol. 28, pp.183–237, 1997. [3] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. New York: NY, North-Holland, 1977. [4] M. H. Lee, “The center weighted Hadamard transform,” IEEE Trans. Circuits Syst., vol. 36, no. 9, pp.1247–1248, 1989. [5] Z. Chen, M. H. Lee, and G. Zeng, “Fast cocyclic Jacket transform,” IEEE Trans. Signal Process., vol. 56, no. 5, pp. 2143–2148, May 2008. [6] G. Zeng and M. H. Lee, “A generalized reverse block Jacket transform,” IEEE Trans. Circuits Syst. I, vol. 55, no. 6, pp. 1589–1600, July 2008. [7] M. H. Lee and G. Zeng, “Fast block Jacket transform based on Pauli matrices,” in Proc. ICC., Scotland, U.K., June 24–28, 2007. [8] G. Feng and M. H. Lee, “An explicit construction of co-cyclic Jacket matrices with any size,” in Proc. Shanghai Conf. Combinatorices, Shanghai, China, May 14–18, 2005. [9] J. Hou and M. H. Lee, “Cocyclic Jacket matrices and its application to cryptography systems,” Lecture Notes in Comput., vol. 3391, pp. 662–668, Jan. 2005.

Xueqin Jiang received the B.S. degree Nanjing Institute of Technology, Nanjing, Jiangsu, P. R. China, in computer science. He received the M.S and Ph.D. degree from Chonbuk National university, Jeonju, Korea, in communication engineering. He is currently working at School of Information Science and Technology, Donghua University, China. His research interests include LDPC codes and coding theory.

ω2 −2ω ω −2ω 22 −2 ω −2 1

ω2 ω −2ω −2ω −2 22 ω 1 −2

−2ω 1 1 −2 ω2 ω2 22 −2ω 2 −2ω 2

ω2 −2ω ω ω −2 1 −2ω 22 −2

ω2 ω −2ω ω 1 −2 −2ω −2 22

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Moon Ho Lee received his B.S. and M.S. degrees in Electrical Engineering from Chonbuk National University, Korea, in 1967 and 1976, respectively, and Ph.D. (I) degree in Electrical Engineering from Chonnam National University in 1984. Also, he received his Ph.D. (II) degree in Information Engineering from the University of Tokyo in 1990. During August 1985 through August 1986, he studied at the University of Minnesota as a Post Doctor. Also, Professional Engineer Licensed at 1981, Korea. Currently, he is a Professor of Division of Electronic and Information Engineering and a Head of the Institute of Information and Communication in Chonbuk National University. His research interests include image processing, mobile communication, high speed communication networks, and information, and algebra coding theory. He is a Member of Sigma Xi. He is a Korea Engineering Academy Member and Guest Editor of IEEE Communication Magazine at 2007, "Quality of service based routing algorithm for heterogeneous networks.”

Ying Guo received the B.S. degree from Qufu Normal University, Qufu, P. R. China, in Mathematics Science in 1999. He received the M.S. degree from Kunming University of Science and Technology, Kunming, P. R. China in Mathematics Science, in 1999. He received the Ph.D. degree Shanghai Jiaotong University, Shanghai, P. R. China, in Communication and Information Science. He is currently working at the Central South University. His research interests include quantum communication and coding theory.

Yier Yan received the B.S. degree from South-Central Nationality University, Wuhan, Hubei, P. R. China, in Applied Electronics. He received the M.S. and Ph.D. degree from Chonbuk National university, Jeonju, Korea, in Communication Engineering. He is currently working at School of Mechanical and Electrical Engineering of Guangzhou University, China. His research interests include information theory, signal processing, and OFDM in MIMO system.

Sarker Md. Abdul Latif received the B.S. (Hons) and M.S. degrees in Applied Physics, Electronics, and Communication Engineering from Islamic University in Bangladesh 2003 and 2004, respectively. He is currently working towards Ph.D. degree on Electronics Engineering from Chonbuk National University, Korea. His research interests include the areas of MIMO, OFDM, wireless communication, Jacket Matrices, UWB, and turbo codes.