On the binary codes with parameters of doubly-shortened 1-perfect codes

Abstract

We show that any binary (n = 2 m − 3, 2 n−m , 3) code C1 is a part of an equitable partition (perfect coloring) {C1, C2, C3, C4} of the n-cube with the parameters ((0, 1, n − 1, 0)(1, 0, n − 1, 0)(1, 1, n − 4, 2)(0, 0, n − 1, 1)). Now the possibility to lengthen the code C1 to a 1-perfect code of length n+2 is equivalent to the possibility to split the part C4 into two distance-3 codes or, equivalently, to the biparticity of the graph of distances 1 and 2 of C4. In any case, C1 is uniquely embeddable in a twofold 1-perfect code of length n + 2 with some structural restrictions, where by a twofold 1-perfect code we mean that any vertex of the space is within radius 1 from exactly two codewords.The hypercube H n = (V (H n ), E(H n )) of dimension n is the graph whose vertices are the all binary n-words, two words being adjacent if and only if they differ in exactly one position.d(·, ·) -the Hamming distance, i.e., the natural graph distance in H n . 0 = 0 . . . 0 (the all-zero word),1 = 1 . . . 1 (the all-one word).A binary code C of length n and code (or minimal) distance d, or (n, |C|, d) code, is a subset of V (H n ) such that d(x,ȳ) ≥ d for any differentx andȳ from C.A partition {C 1 , . . . , C r } of V (H n ) into r nonempty parts is said to be equitable with parameters (S ij ) n i,j=1 if for every i, j ∈ {1, . . . , r} every vertexx from C i has exactly S ij neighbors from C j (the corresponding r-valued function on V (H n ) is known as a perfect coloring).A binary code C ⊂ V (H n ) is said to be 1-perfect if every vertexx ∈ V (H n ) is at the distance 0 or 1 from exactly one codeword. Equivalently, {C, V (H n ) \ C} is an equitable partition with parameters ((0, n)(1, n − 1)). Equivalently, C is a (2 m − 1, 2 2 m −m−1 , 3) code, n = 2 m − 1.We will say that a multiset B ⊂ V (H n ) is a twofold 1-perfect code if every vertexx ∈ V (H n ) is at the distance 0 or 1 from exactly two codewords of B. We will say that a multiset B ⊂ V (H n ) is splittable if it can be represented as the (multiset) union of two distance-3 codes; otherwise B is unsplittable. The existence of unsplittable twofold 1-perfect codes was proved in [6].We say that a code C ′ if obtained by shortening from a code C ⊂ V (H n ) if C ′ = {x ∈ V (H n−1 ) |x0 ∈ C}. Respectively, C ′′ is doubly-shortened from C if 1

On the binary codes with parameters of triply-shortened 1-perfect codes

“…The proofs of (a) for (n, 3) ′′ op (n, 4) ′′ op codes are similar, based on the generated equitable partition [4].…”

Perfect 2‐colorings of Hamming graphs

“…In some cases, optimal, by mean of some bound, objects are proven to be in a one‐to‐one correspondence with perfect colorings with special parameters. For example, unbalanced Boolean functions attaining the correlation‐immunity bound [12]; almost perfect codes and some classes of optimal codes (see, e.g., [27,29]); orthogonal arrays attaining the Bierbrauer–Friedman bound [4,15] induce perfect 2‐colorings of the Hamming graph [40,41]; binary orthogonal arrays attaining the Bierbrauer–Gopalakrishnan–Stinson bound [5] induce perfect 3‐colorings of the hypercube [31]. A very nice result of Potapov [41] shows a one‐to‐one correspondence between the perfect 2‐colorings of the Hamming graph and the Boolean‐valued functions on attaining a bound that connects the correlation immunity of the function, the density of ones, and the average 0–1‐contact number (the number of neighbors with function value 1 for a given vertex with value 0).…”

On $q$-ary shortened-$1$-perfect-like codes

“…Östergård and Pottonen [17] found two (13,512,3) binary codes which are not doublyshortened 1-perfect codes. A generalization on these codes to an arbitrary length of form 2 m − 3 (and also 2 m − 4 for the triply-shortened analog), m ≥ 5, was constructed in [14] (where also all (12,256,3) and (13,512,3) codes were classified up to equivalence). However, it was shown in [12] and [13] that every code with parameters of doubly-or triply-shortened binary Hamming code induces a very regular structure called equitable partition (in the doubly-shortened case, the code is a cell of such partition, while in the triply-shortened extending the code results in a cell of such partition).…”