We solve some first questions in the table of small parameters of completely regular (CR) codes in Hamming graphs$H(n,\ q)$. The most uplifting result is the existence of a {13, 6, 1; 1, 6, 9}-CR code in$H(n,\ 2), n \geq$13. We also establish the non-existence of a {II, 4; 3, 6}-code and a {10, 3; 4, 7}-code in$H(12,2)$and$H(13,2)$. A partition of the complement of the quaternary Hamming code of length 5 into 4-cliques is found, which can be used to construct completely regular codes with covering radius 1 by known constructions. Additionally we discuss the parameters {24, 21, 10; 1,4, 12} of a putative completely regular code in$H(24,2)$and show the nonexistence of such a code in$H(8,4)$.

The Galois ring GR$(4^{\Delta})$is the residue ring$Z_{4}[x]/(h(x))$, where$h(x)$is a basic primitive polynomial of degree$\Delta $over$Z_{4}$. For any odd$\Delta $larger than 1, we construct a partition of GR$(4^{\Delta}) \backslash \{0\}$into 6-subsets of type$\{a,b,-a-b,-a,-b,a+b\}$and 3-subsets of type$\{c,-c,2c\}$such that the partition is invariant under the multiplication by a nonzero element of the Teichmuller set in GR$(4^{\Delta})$and, if$\Delta $is not a multiple of 3, under the action of the automorphism group of GR$(4^{\Delta})$. As a corollary, this implies the existence of quasi-cyclic additive 1-perfect codes of index$(2^{\Delta} -1)$in$D((2^{\Delta} -1)(2^{\Delta} -2)/{6}, 2^{\Delta} -1)$where$D(m,n)$is the Doob metric scheme on$Z^{2m+n}$.

We study codes with parameters of$q$-ary shortened Hamming codes, i.e.,$(n=(q^{m}-q)/(q-1), q^{n-m}, 3)_{q}$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold packings of radius-1 balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal$q$-fold packing with minimum distance 2. Secondly, for every admissible length starting from$n=20$, we show the existence of 4-ary codes with parameters of shortened 1-perfect codes that cannot be obtained by shortening a 1-perfect code.

A λ-fold r-packing (multiple radius- r covering) in a Hamming metric space is a code C such that the radius- r balls centered in the codewords of C cover each vertex of the space by not more (not less, respectively) than λ times. The well-known r-error-correcting codes correspond to the case λ = 1, while in general multifold r-packing are related with list decodable codes. We (a) propose asymptotic bounds for the maximum size of a q-ary 2-fold 1-packing as q grows; (b) prove that a q-ary distance-2 MDS code of length n is an optimal n-fold 1-packing if q ≥ 2n; (c) derive an upper bound for the size of a binary λ-fold 1-packing and a lower bound for the size of a binary multiple radius-1 covering (the last bound allows to update the small-parameters table); (d) classify all optimal binary 2-fold 1-packings up to length 9, in particular, establish the maximum size 96 of a binary 2-fold 1-packing of length 9; (e) prove some properties of 1-perfect unitrades, which are a special case of 2-fold 1-packings.

We solve the problem of existence of perfect codes in the Doob graph. It is shown that 1-perfect codes in the Doob graph D(m, n) exist if and only if 6m + 3n + 1 is a power of 2; that is, if the size of a 1-ball divides the number of vertices.

The dual of the Kasami code of length q2- 1, with q a power of 2, is constructed by concatenating a cyclic MDS code of length q + 1 over Fq with a Simplex code of length q - 1. This yields a new derivation of the weight distribution of the Kasami code, a new description of its coset graph, and a new proof that the Kasami code is completely regular. The automorphism groups of the Kasami code and the related q-ary MDS code are determined. New cyclic completely regular codes over finite fields a power of 2, generalized Kasami codes, are constructed; they have coset graphs isomorphic to that of the Kasami codes. Another wide class of completely regular codes, including additive codes, as well as unrestricted codes, is obtained by combining cosets of the Kasami or generalized Kasami code.

The Doob scheme D(m, n' + n'') is a metric association scheme defined on E4mX F4n'X Z4n'', where E4= GR(42) or, alternatively, on Z42mX Z22n'X Z4n''. We prove the MacWilliams identities connecting the weight distributions of a linear or additive code and its dual. In particular, for each case, we determine the dual scheme, on the same set but with different metric, such that the weight distribution of an additive code C in the Doob scheme D(m, n' +n'') is related by the MacWilliams identities with the weight distribution of the dual code C⊥in the dual scheme. We note that in the case of a linear code C in E4mX F4n', the weight distributions of C and C⊥in the same scheme are also connected.

A {00, 01, 10, 11}-valued function on the vertices of the n-cube is called a t-resilient (n, 2)-function if it has the same number of 00s, 01s, 10s and 11s among the vertices of every subcube of dimension t. The Friedman and Fon-Der-Flaass bounds on the correlation immunity order say that such a function must satisfy t ≤ 2n/3 - 1; moreover, the (2n/3 - 1)-resilient (n, 2)-functions correspond to the equitable partitions of the n-cube with the quotient matrix [[0, r, r, r], [r, 0, r, r], [r, r, 0, r], [r, r, r, 0]], r = n/3. We suggest constructions of such functions and corresponding par titions, show connections with Latin hypercubes and binary 1-perfect codes, characterize the non-full-rank and the reducible functions from the considered class, and discuss the possibility to make a complete characterization of the class.

A λ-fold r-packing in a Hamming metric space is a code C such that the radius-r balls centered in C cover each vertex of the space by not more than λ-times. The well-known r- error-correcting codes correspond to the case λ = 1. We propose asymptotic bounds for q-ary 2-fold 1-packings as q grows, find that the maximum size of a binary 2-fold 1-packing of length 9 is 96, and derive upper bounds for the size of a binary λ-fold 1 -packing.

We show that puncturing the dodecacode, a nonlinear self-dual additive quaternary code of length 12, leads to a uniformly packed code of distance 5. The coset graph of this code is a diameter-3 distance-regular graph on 1024 vertices, with the intersection array {33,30,15;1,2,15}. We discuss the properties of the new distance-regular graph and related quaternary and binary completely-regular codes.

The dodecacode is a nonlinear additive quaternary code of length 12. By puncturing it at any of the twelve coordinates, we obtain a uniformly packed code of distance 5. In particular, this latter code is completely regular but not completely transitive. Its coset graph is distance-regular of diameter three on $$2^{10}$$ vertices, with new intersection array $$\{33,30,15;1,2,15\}$$ . The automorphism groups of the code, and of the graph, are determined. Connecting the vertices at distance two gives a strongly regular graph of (previously known) parameters $$(2^{10}, 495,238, 240)$$ . Another strongly regular graph with the same parameters is constructed on the codewords of the dual code. A non trivial completely regular binary code of length 33 is constructed.

The article “A new distance-regular graph of diameter 3 on 1024 vertices", written by Minjia Shi, Denis S. Krotov and Patrick Solé, was originally published electronically on the publisher's internet portal (currently SpringerLink) on 24 January 2019 without open access.

In this paper, two different Gray-like maps from Zpα× Zpkβ, where p is prime, to Zpn, n = α t βpk-1, denoted by φ and Φ, respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if C is a ZpZpk-additive code, and C⊥is its dual, then the weight enumerators of the image p-ary codes φ(C) and Φ(C⊥) are formally dual. This is a partial generalization of [D. S. Krotov, On Z2k-dual binary codes, IEEE Transactions Information Theory 53 (2007), 1532-1537], and the result is generalized to odd characteristic p and mixed alphabet. In addition, a construction of 1-perfect additive codes in the mixed ZpZp2... Zpk alphabet is given.

A q-ary code of length n, size M, and minimum distanced is called an (n,M,d)q code. An (n,qk,n - k + 1)qcode is called a maximum distance separable (MDS) code. In this paper, some MDS codes over small alphabets are classified. It is shown that every (k + d - 1, qk, d)qcode with k ≥ 3, d ≥ 3, q ∈ (5, 7} is equivalent to a linear code with the same parameters. This implies that the (6, 54, 3)5code and the (n, 7n-2, 3)7MDS codes for n ∈ (6, 7, 8} are unique. The classification of one-error-correcting 8-ary MDS codes is also finished; there are 14, 8, 4, and 4 equivalence classes of (n, 8n-2, 3)8codes for n = 6, 7, 8, and 9, respectively. One of the equivalence classes of perfect (9, 87, 3)8codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction.

A Z2Z4-linear Hadamard code of length α + 2β = 2tis a binary Hadamard code, which is the Gray map image of a Z2Z4-additive code with α binary coordinates and β quaternary coordinates. It is known that there are exactly ⌊t-1/2⌋ and ⌊t/2⌋ nonequivalent Z2Z4-linear Hadamard codes of length 2t, with α = 0 and α ≠ 0, respectively, for all t ≥ 3. In this paper, it is shown that each Z2Z4-linear Hadamard code with α = 0 is equivalent to a Z2Z4-linear Hadamard code with α ≠ 0, so there are only ⌊t/2⌋ nonequivalent Z2Z4-linear Hadamard codes of length 2t. Moreover, the order of the monomial automorphism group for the Z2Z4-additive Hadamard codes and the permutation automorphism group of the corresponding Z2Z4-linear Hadamard codes are given.

Perfect codes obtained by the Vasil'ev-Schönheim construction from a linear base code and quadratic switching functions are transitive and, moreover, propelinear. This gives at least exp(cN2) propelinear 1-perfect codes of length N over an arbitrary finite field, while an upper bound on the number of transitive codes is exp(C(NlnN)2\vphantom)).

Best and Brouwer proved that triply-shortened and doubly-shortened binary Hamming codes (which have length 2m-4 and 2m-3, respectively) are optimal. Properties of such codes are here studied, determining among other things parameters of certain subcodes. A utilization of these properties makes a computer-aided classification of the optimal binary one-error-correcting codes of lengths 12 and 13 possible; there are 237 610 and 117 823 such codes, respectively (with 27 375 and 17 513 inequivalent extensions). This completes the classification of optimal binary one-error-correcting codes for all lengths up to 15. Some properties of the classified codes are further investigated. Finally, it is proved that for any m ≥ 4, there are optimal binary one-error-correcting codes of length 2m-4 and 2m-3 that cannot be lengthened to perfect codes of length 2m-1.

We characterize the set of all n-ary quasigroups of order 4: every n-ary quasigroup of order 4 is permutably reducible or semilinear. Permutable reducibility means that an n-ary quasigroup can be represented as a composition of k-ary and $(n-k+1)$-ary quasigroups for some k from 2 to $n-1$, where the order of arguments in the representation can differ from the original order. The set of semilinear n-ary quasigroups has a characterization in terms of Boolean functions.

A binary poset code of codimensionm(of cardinality 2n-m, wherenis the code length) can correct maximummerrors. All possible poset metrics that allow codes of codimensionmto bem-, (m-1)-, or (m-2)-perfect are described. Some general conditions on a poset which guarantee the nonexistence of perfect poset codes are derived; as examples, we prove the nonexistence ofr-perfect poset codes for somerin the case of the crown poset and in the case of the union of disjoint chains.

We present a construction of 1-perfect binary codes, which gives a new lower bound on the number of such codes. We conjecture that this lower bound is asymptotically tight.

A binary poset code of codimension m (of cardinality 2n-m, where n is the code length) can correct maximum m errors. All possible poset metrics that allow codes of codimension m to be m-, (m-1)- or (m - 2)-perfect are described. Some general conditions on a poset which guarantee the nonexistence of perfect poset codes are derived.

A new generalization of the Gray map is introduced. The new generalization Phi:Z2knrarr Z22k-1n is connected with the known generalized Gray map phi in the following way: if we take two dual linear Z2k-codes and construct binary codes from them using the generalizations phi and Phi of the Gray map, then the weight enumerators of the binary codes obtained will satisfy the MacWilliams identity. The classes of Z2k-linear Hadamard codes and co-Z2k-linear extended 1-perfect codes are described, where co-Z2k-linearity means that the code can be obtained from a linear Z2k-code with the help of the new generalized Gray map