Abstract In two papers entitled “Two generalizations of almost perfect nonlinearity” and “On the vector subspaces of $$\mathbb F_{2^n}$$ F 2 n over which the multiplicative inverse function sums to zero”, the first author has introduced and studied the notion of sum-freedom of vectorial functions, which expresses that a function sums to nonzero values over all affine subspaces of $$\mathbb {F}_{2^n}$$ F 2 n of a given dimension $$k\ge 2$$ k ≥ 2 , and he then focused on the k th order sum-freedom of the multiplicative inverse function $$x\in \mathbb {F}_{2^n}\mapsto x^{2^n-2}$$ x ∈ F 2 n ↦ x 2 n - 2 . Some general results were given for this function (in particular, the case of affine spaces that do not contain 0 was solved positively), and the cases of $$k\in \{3,4,n-4,n-3\}$$ k ∈ { 3 , 4 , n - 4 , n - 3 } and of k not co-prime with n were solved as well (negatively); but the cases of those linear subspaces of dimension $$k\in \llbracket 5;n-5\rrbracket $$ k ∈ 〚 5 ; n - 5 〛 , co-prime with n , were left open. The present paper is a continuation of the previous work. After studying, from two different angles, the particular case of those linear subspaces that are stable under the Frobenius automorphism, we deduce from the second approach that, for k small enough (approximately, $$3\le k\le n/10$$ 3 ≤ k ≤ n / 10 ), the multiplicative inverse function is not k th order sum-free. Finally, we deduce from results previously obtained in the second paper mentioned above, that for any even n and every $$2\le k\le n-2$$ 2 ≤ k ≤ n - 2 , the multiplicative inverse function is not k th order sum-free.

Let Rq(r,n) be the rth order q-ary Reed-Muller code of length qn, which is the set of functions from \mathbb Fqnto \mathbb Fqrepresented by polynomials of degree ≤ r in \mathbb Fq[X1, ... ,Xn]. The affine linear group AGL(n,\mathbb Fq) acts naturally on Rq(r,n). We derive two formulas concerning the number of orbits of this action: (i) an explicit formula for the number of AGL orbits of Rq(n(q-1),n), and (ii) an asymptotic formula for the number of AGL orbits of R2(n,n)/R2(1,n). The number of AGL orbits of R2(n,n) has been numerically computed by several authors for n ≤ 31; the binary case of result (i) is a theoretic solution to the question. Result (ii) answers a question by MacWilliams and Sloane.

In this paper we report the following computational results on partial spread functions in eight variables: (i) the numbers of equivalence classes of partial spread functions (in eight variables) of all possible orders; (ii) the total number of partial spread bent functions in eight variables; (iii) the distribution of the cardinalities of stabilizers (in GL(8, F2)) of partial spread bent functions in eight variables. The computational method is also described.

Let Psik,ndenote the number of inequivalent binary self-orthogonal [n, k] codes. We present a method which allows us to compute Psik,nexplicitly for a moderate k and an arbitrary n. Included in this paper are explicit formulas for Psik,nwith k les 5.

A longstanding conjecture by Niho on the maximally nonlinearity of certain power functions was proved recently by Hollmann and Xiang using a result of Dobbertin on the almost perfect nonlinearity of the Niho power functions. A key ingredient of the proof, a bound for certain binary weights, was obtained using a computer. In this note, we provide a noncomputer proof for the bound of the binary weights.

After a discussion of automorphisms of Reed-Muller codes the authors show that the Reed-Muller code R(r,m) is not Z/sub 4/-linear for 3/spl les/r/spl les/m-2, proving a conjecture by Hammons, Kumar, Calderbank, Sloane, and Sole (1994).

Let /spl rho/(1,m) and N(1,m) be the covering radius and norm of the first-order Reed-Muller code R(1,m), respectively. It is known that /spl rho/(1,2k+1)/spl les/lower bound [2/sup 2k/-2/sup (2k-1/2)/] and N(1,2k+1)/spl les/2 lower bound [2/sup 2k/-2/sup (2k-1/2)/] (k>0). We prove that /spl rho/(1,2k+1)/spl les/2 lower bound [2/sup 2k-1/-2/sup (2k-3/2)/] and N(1,2k+1)/spl les/4 lower bound [2/sup 2k-1/-2/sup (2k-3/2)/] (k>0). We also discuss the connections of the two new bounds with other coding theoretic problems.

We prove that the covering radius of R(1, 11) in R(3, 11) is 992, and that the covering radius of R(1, 13) in R(3, 13) is 4032, both not exceeding the quadratic bound.

Let R(r,m) be the rth-order Reed-Muller code of length 2/sup m/ and let rho (r,m) be its covering radius. R

Whether quasi-perfect codes are normal is addressed. Let C be a code of length n, dimension k, covering radius R, and minimal distance d. It is proved that C is normal if d>or=2R-1. Hence all quasi-perfect codes are normal. Consequently, any (n,k)R binary linear code with minimal distance d>or=2R-1 is normal.>

Let C be a binary linear code with covering radius R, and C/sub 0/ a subcode of C of codimension i. An upper bound is obtained for the covering radius of C/sub 0/ in terms of R and i. When C/sub 0/ =(0), the bound becomes the sphere covering bound for R.>

Two upper bounds for the norm N(C) of a binary linear code C with minimal weight d and covering radius R are given. The second of these bounds implies that C is normal if R=3.>

Some new lower bounds on mod C mod for a binary linear (n, k)R code C with n+1=t(R+1)-r(02 odd) or with n+1=t(R+1)-1(t>2 even) are obtained. These bounds improve the sphere covering bound considerably and give several new values and lower bounds for the function t(n, k), the smallest covering radius of any (n, k) code.>

Let C be a binary code (not necessarily linear) with covering radius R and length n=3R+2. The sphere covering bound on the cardinality of C is improved considerably provided C has minimal distance d>2. Some new results on the function t(n,k) (the smallest covering radius of any binary linear code with length n and dimension k): t(38.6)>or=13, t(47.7)>or=16, t(59.8)>or=20 are given.>