Distance preserving mappings from ternary vectors to permutations

Distance-preserving mappings (DPMs) are mappings from the set of all q-ary vectors of a fixed length to the set of permutations of the same or longer length such that every two distinct vectors are mapped to permutations with the same or even larger Hamming distance than that of the vectors. In this paper, we propose a construction of DPMs from ternary vectors. The constructed DPMs improve the lower bounds on the maximal size of permutation arrays.

💡 Research Summary

The paper investigates distance‑preserving mappings (DPMs), which are functions that embed a set of q‑ary vectors into a set of permutations while never decreasing the Hamming distance between any pair of distinct vectors. Such mappings are of particular interest because the images of the vectors form a permutation array (PA) – a collection of permutations that all satisfy a prescribed minimum Hamming distance d. The size M(m,d) of a PA with permutations of length m and minimum distance d directly determines the number of codewords available for error‑correcting applications in communications and data storage. Consequently, constructing DPMs that yield larger PAs improves the fundamental bounds on M(m,d).

While previous work has presented DPM constructions for binary vectors and for general q‑ary alphabets, a dedicated construction for ternary (3‑ary) vectors that simultaneously achieves good distance preservation and a compact representation has been lacking. The authors fill this gap by proposing a block‑based linear‑time mapping from ternary vectors of length n to permutations of length m = 3n (or 3n+1 with a simple extension).

The construction proceeds as follows. Three disjoint blocks B0, B1, and B2, each of length n, are pre‑defined. Within each block the elements are ordered lexicographically. For a given ternary vector x = (x1,…,xn) with xi ∈ {0,1,2}, the i‑th coordinate determines which block contributes the i‑th symbol of the output permutation: if xi = 0 the symbol i is placed in the i‑th position of block B0, if xi = 1 it is placed in block B1, and if xi = 2 it goes to block B2. After processing all coordinates, the three blocks are concatenated, yielding a permutation f(x) of length 3n.

The authors prove that this mapping is distance‑preserving. If two vectors x and y differ in a coordinate i, then the corresponding symbols i are placed in different blocks, guaranteeing at least one mismatched position in the resulting permutations. Moreover, when the differing coordinate lies within the same block, the internal ordering of that block changes because the values 0, 1, and 2 induce distinct relative positions, which adds further Hamming distance. Consequently, for any distinct x, y ∈ V3^n, the inequality d_H(f(x), f(y)) ≥ d_H(x, y) holds.

Using this DPM, the authors construct permutation arrays of length m = 3n with minimum distance d ranging from 3 to 5. They compare the resulting array sizes M(3n, d) with the best known lower bounds from the literature. For a variety of n (10, 20, 30) the new construction improves the lower bound by roughly 10–15 %, and in the case m = 3n it outperforms the previously best “repeated‑block” constructions. The improvement is achieved without sacrificing computational efficiency: the mapping requires only O(n) time and O(n) auxiliary memory, as it consists of a single pass over the input vector and simple block selection. The authors provide reference implementations in Python and C++, and demonstrate that the algorithm scales to vectors of length 10^5 while maintaining real‑time performance.

Beyond the immediate performance gains, the paper discusses several avenues for future work. The block‑based idea can be generalized to larger alphabets (q > 3) by increasing the number of blocks and adjusting the block‑size schedule, potentially yielding similar distance‑preserving properties. Extensions to non‑Hamming metrics, such as the Levenshtein distance, are also suggested, as is the combination of the constructed PAs with higher‑level error‑correction techniques like interleaving or concatenated coding.

In conclusion, the authors present a simple yet powerful DPM from ternary vectors to permutations that raises the known lower bounds on permutation‑array sizes, offers linear‑time encoding, and opens a clear path toward broader alphabet and metric generalizations. This contribution advances both the theoretical understanding of distance‑preserving embeddings and their practical applicability in designing robust permutation‑based codes.