An extremal type I self-dual code of length 16 over 𝔽2+u𝔽2

Recently, a comprehensive examination of self-dual codes over the alphabet lF2 + ulF2 was published. This included a classification of all self-dual codes up to length 8, and tables of extremal codes up to length 36 for Type I codes and length 40 for Type II codes. Explicit constructions were given except for the Type I code of length 16. A construction for this code is given here. The introduction of codes over Z4 and their connection with nonlinear binary codes [8] and unimodular lattices [3, 7] has created tremendous interest in codes over rings. Another alphabet of size 4, lF2 + ulF2, was used in [1] to construct lattices. Codes over IF 2 + ulF 2 have also been used in the construction of self-dual binary codes [7] and formally self-dual binary codes [2]. A linear code 0 over lF2+ulF2 oflength n is an lF2+ulF2-submodule of (lF2+ulF2)n. The Lee weight wdx) of x = (Xl, X2, ... , xn) is defined as nl (x) + 2n2(X) where no(x) is the number of Xi = 0, n2(x) the number of Xi = u and nl(x) = n no(x) n2(x). The Lee distance dL(x, y) between two codewords X and Y is the Lee weight wtL(X-Y) of x y. The minimum Lee weight dL of 0 is the smallest Lee weight among all non-zero codewords of O. Define the inner-product x . Y of X = (Xl, X2,' .. , xn) and Y = (Yl, Y2,·· ., Yn) in (lF2 + ulF2)n by XIYl + X2Y2 + ... + XnYn' The dual code OJ... of 0 is defined as {x E (lF2 + ulF2 )n I X· Y = 0 for all YEO}. 0 is said to be self-dual if 0 = OJ.... A self-dual code over IF 2 + ulF 2 is said to be Type II if the Lee weight of every codeword is a multiple of 4 and Type I otherwise. Australasian Journal of Combinatorics 19(1999), pp.235-238 Corollary 1.1 ([5]) Let dL(II, n) and dL(I, n) be the highest minimum Lee weights of a Type II code and a Type I code, respectively, of length n. Then dL(II,n) < 4 [~] + 4, { 2 [2"+6] if n =1= 1,6,11,16, dL(I, n) < 10 ' 2 [2~!6] + 2, otherwise. Codes which meet these bounds are called extremal. For this ring, the Gray map is defined in [1] as follows: ¢> : ((JF2 + uJF2 )n, Lee distance) -t (JF~n, Hamming distance) where ¢>(x + uy) = (y,x + y) for an element x + uy E (JF2 + uJF2)n,x,y E JF~. The Gray map is a distance preserving map. Proposition 1.2 ([2]) The image of the Gray map of a self-dual code Cover JF 2 + uJF 2 is a self-dual binary code. The minimum Lee weight of C is the same as the minimum weight of ¢>( C). 2 Double Circulant Codes A pure double circulant code of length 2n has a generator matrix of the form ( I , R) where I is the identity matrix of order nand R is an n by n circulant matrix. A code with a generator matrix of the form