DPMs from Ternary Vectors
5.2 Future Works
Distance-preserving mappings are well studied in the past few years, especially binary distance-preserving mappings. However, there are still a lot of topics worth exploring.
z An initial mapping for the construction of ternary n_(n+1)-DIMs.
If a DIM from Z3m to Sm+1 can be found, then ternary n_(n+1)-DIMs for all n≥ m can be constructed by using Construction E in Section 4.1.
z Constructions of ternary n-DIMs.
Construction E can only construct ternary n-DPMs even though the initial mapping is distance increasing. Thus, a new construction method must be found.
z A non-recursive systematic construction method of (n, δ)-DIMs from binary or ternary vectors.
A mapping that increases more distance than that of input vectors may be more interesting for applications. Although Chang has proposed constructions of binary (n, δ)-DIMs for δ ≥ 2 [13], the constructions are recursive and are unable to construct binary (n, δ)-DIMs for all n > nδ where nδ is the smallest positive integer nδ such that a binary (n, δ)-DIM could be constructed for any n ≥ nδ.The concept can also be applied to ternary DIMs. No ternary (n, δ)-DIM has been found for δ ≥ 1.
z A tighter bound on the smallest positive integer nδ such that a binary (n, δ)-DIM exists for all n > nδ.
For any δ ≥ 2, Chang provided a bound on the smallest positive integer nδ by which a binary (n, δ)-DIM can be constructed for all n > nδ [13]. It seems that
Chapter 5 Conclusions 76
the bound can be further improved.
z A general construction method of DPMs or DIMs from Zqn to SN.
DPMs and DIMs from Zqn to SN are useful in converting a q-ary code, e.g.
Reed-Solomon code, to a permutation array. The constructions of DPMs or DIMs proposed so far are from binary or ternary vectors and can not be generalized to q-ary vectors for all q ≥ 2. It is a great contribution if one can propose a general construction of DPMs or DIMs from q-ary vectors for all q≥ 2. If the construction is recursive, an initial mapping should also be provided.
z Construct DPMs or DIMs from a subset of vectors.
In present DPMs and DIMs, all vectors are considered. The distance between each pair of vectors in the vector space should be preserved or even increased.
However, this is somewhat overkill. In constructing permutation arrays from distance-preserving mappings, only a subset of vectors, which form a (n, d) code, is mapped to permutations, whereas the other vectors are irrelevant.
Therefore, only the subset of vectors should be either distance preserving or distance increasing.
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