A k-cube graph construction for mappings from Binary vectors to permutations

Abstract

A new graph theoretic construction mapping binary sequences to permutation sequences is presented. The k-cube graph construction has reached the upper bound on the sum of the distances for certain values of the length of the permutation sequence. This contributed in a better way to understand the distance-reducing mapping, which was not investigated before.

Mathematical Approach and Implementation of Frequency Mapping Techniques in Power-Line Communications Channel

“…The total distances, D and E play a role in the permutation code's error correcting capabilities, as was shown in [15]. In the case of M = 4 as depicted in Example 6.2 in the Appendix, the mapping can increase the distance and therefore the error correction capability of the resultant code.…”

A Graph Theoretic Approach for Certain Properties of Spectral Null Codes

“…6 shows the starting permutation sequence as 135. We swap the state-symbol with the following state-symbol in the permutation sequence based on the k-cube construction [12]. We end the swapping process at the last state in the graph.…”

Permutation sequences and coded PAM signals with spectral nulls at rational submultiples of the symbol frequency

“…To create mappings, we make use of the cube graph construction [14] in our work, because it is the only simple construction to our knowledge to provide the highest sum on the Hamming distances, which plays a role in the error correcting capabilities, compared to other constructions [6][7][8]. We use N k (M, n, δ) to denote the SN-DPMs from binary sequences of length n to permutation sequences of length M with spectral nulls at multiples of 1/k.…”

“…The finite state machine of the resultant PAM signals will have the same finite state machine of the base code [14], which in this case is the convolutional code with K = 3 and R = 1/2.…”