سوالهای باز
Open problems posed in [IEEE Trans. Inform. Theory, ۶۱ (۲۰۱۵), ۳۲۴۱-۳۲۵۰.]
Preliminary definitions (PDF)
Problem (۱): Prove the nonexistence of perfect codes in S_n, using the Kendall τ-metric, for more values of n and/or other distances
Articles related to the Problem (۱)
-metric. Des. Codes Cryptogr. ۸۹, ۲۵۱۱–۲۵۳۱ (۲۰۲۱)[۱] Wang, X., Wang, Y., Yin, W. et al. Nonexistence of perfect permutation codes under the Kendall τ
[۲] A. Abdollahi, J. Bagherian, F. Jafari, M. Khatami, F.Parvaresh and R. Sobhani, New Bounds on the Size of Permutation Codes With Minimum Kendall T-distance of Three
Problem (۲): What is the size of an optimal anticode in S_n with diameter D
Problem (۳): Improve the lower bounds on the sizes of codes in S_n with even minimum Kendall τ-distance
Problem (۴): Can the codes in S_۵ and S_۷ from Section V be generalized for higher values of n and to larger distances? Are these codes of optimal size
Articles related to the Problem (۴)
Note that the auther of the latter paper formulated the calculation of P(n, d) as a binary integer program with linear/quadratic constraints and found its value
for n ∈ {۵, ۶}. By using a solver (Gurobi Optimization Inc), the values of P(۵, d) for d ∈ {۳, ۴, ۵, ۶} and P(۶, d) for d ∈ {۴, ۵, . . . , ۱۰} has been found successfully . But the solver has not been able to find the optimal value of P(۶, ۳) even after several weeks of computation. The solver has found a heuristic solution of cardinality ۱۰۲ improving the lower bound on P(۶, ۳) from the previous value of ۹۰. Click here to see ۷۲۰ constraints of the optimization problem corresponding to P(۶,۳).
آخرین بروزرسانی:
1404/04/16 - 11:48:37