Block Coded Modulation Using Two-Level Group Codes Over Generalized Quaternion Groups

This paper deals with Euclidean space codes obtained from two-level group codes of length n over a Generalized Quaternion Group Q,, m23, with 2"' elements, with generators x and y, given by the presentation Q,,,= using a matched labelling of a 4-dimensional signal set with 2" elements with Q,. I. TWO LEVEL GROUP CODES OVER e,. Four dimensional signal sets in connection with Q, have been studied in [1,2]. The block diagram of the two-level Block Coded Modulation scheme over Q,, discussed in this paper is shown in Fig. 1. a signal —-> point in 4n dimen sions. Fig. 1 Block diagram of a Twolevel construction S is a four dimensional signal set i -> 5 = (cos(k9),sin(kO),O,O), (O,O,cos(kO),-sin(kO)), Elements of S are indexed (labelled) with elements of Q, by the mapping p given by p(x) = (cos(/&), sin(k6), 0,O) p(yx) = (O,O,cos(k9),-sin(k6)), 05k<2'"-'. It is easily verified that p is a matched labelling [3]. C,, and C, are length n codes over alphabets & = {0,1}. and 2L,_, = (0, I, . . . , 2 " ' — 1). We define a one-one mapping ~:Z~CZ~~-~-+Q,,, as f(a,b) = y"x". (1) For codewords <^ = (an,at, ...,U,,-,) E Cy and -b=(bo,b,,...,b,-,)E C,, the pair (q,b, ), i = 0 , 1 , . . . , n 1 , is identified with the elementf(q, b,) = y 'x ' E Q,, and this associates with the pair (g,b) an n -tuple over Q,, denoted by G,~The subset {c&gE; Cy,b E Q of Q:, denoted by c c x y \ is called the two-level code over Q, with component c c codes C, and C,. If x 'y v is a subgroup of Q under componentwise operation then it is called a two-level group code. c c. Theorem 1: The two-level codex 'y y is a group code over Q, if and only if (i) C, and C, are linear codes over Z, and Z „., respectively, and (ii) C, 0 (2 , 'Cy © 2C,) c C,, where 0 denotes componentwise integer multiplication and 0 denotes componentwise addition modulo 2,'. Notice that Q, is not a semi-direct product group and hence Theorem 1 can not be obtained from the algebraic conditions given in [4] for semi-direct product groups. 11. MSED OF SIGNAL SPACE CODES The mapping f given by (I), is shown below in the tabular form: z,\z,..,