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By Bruno Poizat

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19). 20). Now let u E (L(CW . 1), we can write u = 8 c + v for some c E 71,~2t , V E L. 20); so U E L(C). CHAPTER 2. SELF-DUAL LATTICES 46 Finally we assume that C ::) ~[n, mJ and L(C) is self-dual. 19). Then c E C, because C/~[n,mJ ~ o L(C)/L. 22) These two steps constitute the gluing procedure of our untwisted gluing technique. 5. Assumptions and settings are the same as in the above. ); 8 u ; ; 8"j ;8 wj; i E l,1, jE 1,1') is a U-shell of self-duallattices. 0 (C; f; Ui; Vj; Wj; i E l,1, j E Next we give a decomposability theorem of construction in an important special case.

Let C k be a confe rence matrix. (1) Th e matrix (h , C k ) gen erates a self-dual code of length 2k over Zn when k == o (mod n). (2) 1f n = 2m and m 2 + k == 0 (mod n) , then (h , C k + mh ) generat es a self-dual code of length 2k over Zn. 0 Next we study the relations between self-dual eodes over Zn for different n . Let m , n , I be positive integers and n = Im 2 • Set Ynk . 11) ~ Z by 'fJn (Q) = j if Q == j < n. If we view an integer m as an element of Zn, then 'fJn(m) is the non- Throughout this book , we define the seetion map 'fJn : Zn in Zn for 0 ~ j negative residue of m with respeet to n.

The situation now is much more complicated than that in the previous section. ); Xi; (j; ~j; Wj y; i E T,S, jE 1,t) be a T-shell of self-duallattices. 2) . 17-18) . 4. Let C be a code of length (s + 2t) over ZM' A vector Y(C) E Z~2t and a map 'l/Jc : C --+ Z are called the admissible vector and map of C, respectively if for all cE C. 8) Ir such Y(C) and 'l/Jc exist, then we say that C is admissible to the T-shell 8. In the case that C is admissible to 8, let t be an integral variable, and we call J[8,C; t] = (W a twist factor of C with respect to with respect to 8 if + ey(C) + ty, W + ey(C) + ty) 8.