Abstract We compute the weight distribution of the binary Reed–Muller code $${\mathcal {R}} (4,9)$$ R ( 4 , 9 ) by combining the methodology described in D. V. Sarwate’s Ph.D. thesis from 1973 with newer results on the affine equivalence classification of Boolean functions. More specifically, to address this problem posed, e.g., in the book of MacWilliams and Sloane, we apply an enhanced approach based on the classification of Boolean quartic forms in eight variables due to Ph. Langevin and G. Leander, and the recent results on classification of the quotient space $${\mathcal {R}} (4,7)/{\mathcal {R}} (2,7)$$ R ( 4 , 7 ) / R ( 2 , 7 ) obtained by V. Gillot and Ph. Langevin.

In 2001 Lee, Rajan and Park introduced (among other things) the concept of extended complex reverse jacket (CRJ) transform over finite field. In this paper, we prove that this transform is equivalent to Galois field Fourier (GFF) transform and possesses an additional feature of having a matrix with specific border. We also, present examples of Jacket transforms over finite fields applicable to odd dimensions extending in this way previous works on the subject.

In this paper, an amplification of the Cusick-Cheon conjecture on balanced Boolean functions in the cosets of the binary Reed-Muller code RM(k,m) of order k and length 2m, in the cases where k = 1 or k ges (m-1)/2, is proved.

The average conditional entropy of the key given the message and its corresponding cryptogram, ff(K|M, C), which is refer as a key appearance equivocation, was proposed as a theoretical measure of the strength of the cipher system under a known plaintext attack by Dunham in 1980. In the same work (among other things), lower and upper bounds for H(SM |MLCL) are found and its asymptotic behavior as a function of cryptogram length L is described for simple substitution ciphers, i.e., when the key space SLambda1, Is the symmetric group acting on a discrete alphabet M. In the present paper we consider the same problem when the key space is an arbitrary subgroup KMand generalize Dunham's result.

Carlet and Charpin classified the set of cubic (n-4)-resilient Boolean functions into four different types with respect to the Walsh spectrum and the dimension of the linear space. Based on the classification of RM(3,6)/RM(1,6), we have completed this classification of cubic (n-4)-resilient Boolean functions by deriving the corresponding algebraic normal form (ANF) and autocorrelation spectrum for each of the four types. At the same time, we have solved an open problem by proving that all plateaued cubic (n-4)-resilient Boolean functions have dimension of the linear space equal either to n-5 or n-6.

Let R/sub t,n/ be the set of t-resilient Boolean functions in n variables, and let /spl rho//spl circ/(t,r,n) be the maximum distance between t-resilient functions and the rth-order Reed-Muller code RM(r,n). We prove that /spl rho//spl circ/(t,2,6)=16 for t=0,1,2 and /spl rho//spl circ/(3,2,7)=32, from which we derive the lower bound /spl rho//spl circ/(t,2,n) /spl ges/ 2/sup n-2/ with t /spl les/ n-4. Using a result from coding theory on the covering radius of (n-3)th- and (n-4)th-order Reed-Muller codes, we establish exact values of the covering radius of RM(n-3,n) in the set of 1-resilient Boolean functions in n variables, when /spl lfloor/n/2/spl rfloor/=1 mod 2 and lower bounds of RM(n-4,n) in the set of 2-resilient Boolean functions in n variables. This result leads again to different lower bounds for general dimensions n and r=0 or 3 mod 4.