Title:

Describing quaternary codes using binary codes

For a quaternary code C of length n, de ne a pair of binary codes fC1;C2g as: C1 = C mod 2 C2 = h(C \ 2Zn 4 ) where h is a bijection from 2Z4 to Z2 mapping 0 to 0 and 2 to 1 and for the extension to a map acting coordinatewise. Here C1 C2. For a pair of binary codes fC1;C2g with C1 C2, let C(C1;C2) be the set of Z4codes giving rise to this binary pair as de ned above. Our main goal is to describe the set C(C1;C2) using the binary pair of codes fC1;C2g. In Chapter 1, we give some preliminaries. In Chapter 2, we start with a general description of codes fC1;C2g which give cardinality of C(C1;C2). Then we show that C(C1;C2) ' C 1 Zn 2 =C2. The cohomology of C(C1;C2) is given in Section (2:2). Then we end chapter 2 with a description of dual codes of C(C1;C2). Chapter 3 is about weight enumerators of codes in C(C1;C2). The average swe is given in terms of weight enumerators of C1 and C2 in Section(3:1) as swe(x; y; z) = jC2j 2n (weC1(x + z; 2y) (x + z)n) + weC2(x; z) Detailed computations of swe's of codes in C(C1;C2) using codes fC1;C2g is then given. Information about di erent weight enumerators of codes in C(C1;C2) is given in Section (3:2). These weight enumerators are included in an a ne space of polynomials. Then we end chapter 3 with a description of weight enumerators of self dual codes. Chapter 4 deals with actions of 2 the automorphism group G = Aut(C1) \ Aut(C2) Sn on C(C1;C2) which preserves cwe of codes. Corresponding action on C 1 Zn 2 =C2 is explained in this chapter. Changing signs of coordinates can be de ned as an action of Zn 2 on C(C1;C2). This action preserves swe of codes. Corresponding action on C 1 Zn 2 =C2 is provided in this chapter. In the appendix, we give a complete description of Z4codes in C(C1;C2) with C1 = C2 = Extended Hamming Code of length 8. A programming code in GAP for computing derivations is given. And a description of the a ne space containing the swe's of Z4codes is given with examples of di erent C1 = C2 having same weight enumerator.
