1345
1) The diagram
H { depicts a partition of F; into affine hyperplanes H, H' such that
I H n S I = a and I H ' n S I = B .
Three points of S add up to a point of H if and only if Hcontains an odd number of these three points. Thus Property * yields the inequality ( ; ) + a ( 'I) 2 1 H I -a = 2 8 -a .2) In the same way, the diagram WI + L depicts a partition of F; into four two-codimensional affine subspaces. (The Greek letters denote the cardinalities of their intersections with S.) Note that the union of any two of these subspaces is an affine hyperplane of F;.Three points of S add up to a point of the subspace L if and only if each of the three hyperplanes through L contains an odd number of these points. Hence Property * implies the inequality Having now dealt with preliminaries, we proceed to prove the 1) First, A , = 0, i.e., all elements of S are distinct. nonexistence of C in five steps.Proof: Assuming the contrary we get a contradiction by virtue of Property * :( J ) < 2 9 -n for n < 1 6 .
2)Next we have B, = B, = B3 = 0. Proof: If not, the affine hyperplane that intersects S in the particular codeword would be deficient with respect to Property *: n 2 . ( y ) < 2 8 -2 U 3) Two words of weight 12 in C L cannot intersect in a 9 set. Proof: If the two words were cut out by hyperplanes H, and H,, then the seven-dimensional affine space F:\( H, U H 2 ) would be deficient with respect to Property * : H,{ l . ( ( ;)+( ; I + ( : ) ) + 3 . 3 . 9 < 2 7 -1 . 3 1 1 --H2 4 ) We have A,, 2 Blo( = B,).Proof: We show that each codeword of C of weight 10 is also a codeword of C. Suppose a hyperplane H intersects S in a 10 set T : = H f~ S. Then T spans the eight-dimensional affine space H , for otherwise a partition of the type would exist, in contradiction with 2).Hence T contains just one word of C. If this word were properly contained in T, one of the points of T would be affinely independent of the nine remaining ones. This would imply the existence of a partition by 2) and 3). 5) We apply the MacWilliams identities to show that l ) , 2), and 4 ) conflict. The following equations are relevant for our purpose: 2 (We have taken into account that C is an even code and that A , = B, = B, = B3 = 0.) Elimination enables us to express A,, and Blo( = $) in terms of A,, A,,, and A16: A,, = A, + 5A1, + 16A,, -16 B,, = 4A, + 12A14 + 160A16 + 32. A,, 2 B,,, which we derived in step 4). Hence C does not exist.
This contradicts the inequality
CONCLUDING REMARKBy similar methods, one can, for instance, prove that t [16,9] = 3 and t[23,15] = 3. In a future paper we intend to present our seemingly ad hoc techniques in a more generally applicable form. Abstract -A close relationship between the weight enumerators of two cyclic codes is presented. The weights of the codewords are expressed in terms of Kloosterman sums. The Weil-Carlih-Uchiyama estimate for Kloosterman sums is used to obtain bounds for the minimum distance of the codes. A relation with negacyclic codes is discussed. K ( A ; a, b ) is defined by K ( A ; a , b ) = A(a...
