Conclusion and Future Works
Definition 7.5.1. Given a code C and a metric δ, determine whether the covering radius of C under δ is less than or equal to some value b
Similar to what we mentioned in Chapter 6, when the problem instance is of sizeO (log |C|), the naive algorithm for covering radius runs inefficiently in exponential time. Moreover, the decision problem for the covering radius problem of a binary linear code under Ham-ming distance is ΠP2-complete, see McLoughlin’s work [29]. McLoughlin actually gave a
polynomial-time reduction from the AE qualified 3-Dimensional matching problem. But we cannot directly apply this reduction to the covering radius problem of a permutation code, since permutation codes have some different nature from the linear codes.
Definition 7.5.2. (CRSPAδ) Given a code C and a metric δ, determine whether the covering radius of C under δ is less than or equal to some value b.
Conjecture 7.5.3. CRSPAℓ∞ is ΠP2-complete.
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Appendix A
Tables of Ball Size
We list the tables of ball size which are not given in previous results.
Table A.1: The table of ball size under ℓ∞-metric for λ = 2, m∈ [20], d = 2 n V (2, n, 2, ℓ∞)
2 1
4 6
6 90
8 786
10 6139
12 54073
14 477228
16 4113864
18 35579076
20 308945881 22 2679325561 24 23222971098 26 201351085146 28 1745886520422 30 15137227297027 32 131243141767393 34 1137923361184848 36 9866167034815440 38 85542686564024352 40 741681846818742097
Table A.2: The table of ball size under ℓ∞-metric for λ = 2, m∈ [20], d = 3 n V (2, n, 3, ℓ∞)
2 1
4 6
6 90
8 2520
10 45450
12 669666
14 9747523
16 154700569
18 2502207156
20 40043708244
22 632349938520 24 9986116318524 26 158192179607364 28 2509767675626581 30 39796612230719845 32 630688880128338378 34 9994168619297530758 36 158396161513685960664 38 2510580301930785916566 40 39792149406721332018414
Table A.3: The table of ball size under ℓ∞-metric for λ = 2, m∈ [20], d = 4 n V (2, n, 4, ℓ∞)
2 1
4 6
6 90
8 2520
10 113400
12 3540600
14 88610850
16 2044242426
18 47806940971
20 1196081134201
22 30647443460124 24 784921116539484 26 19899840884886720 28 500019936693729120 30 12551808236761063440 32 315694279415609776404 34 7955400980632212027852 36 200622722060793477132937 38 5057787000067792980984649 40 127452627155747602225756890
Table A.4: The table of ball size under ℓ∞-metric for λ = 2, m∈ [20], d = 5 n V (2, n, 5, ℓ∞)
2 1
4 6
6 90
8 2520
10 113400
12 7484400
14 361859400
16 14091630840
18 489147860970
20 16420511188146
22 563209318269379
24 20416518083009593 26 758713036253909844 28 28351365170599079604 30 1054143198114097909680 32 38864351069181445164480 34 1423417411123883479886400 36 52064892889568503574209920 38 1906534315066176639758670480 40 69931615009402042606373019804
Table A.5: The table of ball size under ℓ∞-metric for λ = 3, m∈ [20], d = 2 n V (3, n, 2, ℓ∞)
3 1
6 20
9 1680
12 61340
15 1886431
18 69496201
21 2568223000
24 91712960320
27 3290467596440
30 118724053748417
33 4276273204804217 36 153904262366842444 39 5541519231941145440 42 199545071017172522244 45 7184755645113714298863 48 258691998154725997048673 51 9314545233907934721851472 54 335381528796576643131475840 57 12075785123501322139824319056 60 434802491356562053648077727185
Table A.6: The table of ball size under ℓ∞-metric for λ = 3, m∈ [20], d = 3
n V (3, n, 3, ℓ∞)
3 1
6 20
9 1680
12 369600
15 41480880
18 3422150780
21 276888204387
24 25512718688405
27 2418264595619240
30 225661997838758560
33 20649533952628896000 36 1889648253594082624960 39 173699198403114756474600 42 16001577154624484682748453 45 1472965856766989578006355117 48 135481185586476496195656612044 51 12459839493182349378716705969200 54 1146141579672729885487800599057600 57 105440511941055519854115528116882480 60 9699923367172090411762252385134967844
Table A.7: The table of ball size under ℓ∞-metric for λ = 4, m∈ [20], d = 2
n V (4, n, 2, ℓ∞)
4 1
8 70
12 34650
16 5562130
20 708212251
24 114774147001
28 18679465660540
32 2906167849870600
36 454904037056013460
40 71729455730285511001
44 11285129375761977675001 48 1773699532985462649188410 52 278931562239767189408085850 56 43869015908453746845566145990 60 6898693708786029238293860809251 64 1084865341390442288732669957148001 68 170605963060816377946936433265175680 72 26829411396875692269491197638918648400 76 4219165662049303123773116859323196816720 80 663502408038018748448058464247159216890001
Table A.8: The table of ball size under ℓ∞-metric for λ = 5, m∈ [20], d = 2
n V (5, n, 2, ℓ∞)
5 1
10 252
15 756756
20 549676764
25 298227062281
30 218838390759073
35 161446400503248672
40 112632613848657302400
45 79169699996993643966432
50 56151546386557366024202177
55 39717291593245217794329362081
60 28058660061656964336359435570604 65 19835819533825566529982592591911412 70 14024417724324420598672399947721245804 75 9914206081036463014882722168252570938889 80 7008596284293402975749309111124669929079521 85 4954676885097638926007640423100194180529855296 90 3502659589845301193905028251874353899223998638208 95 2476160267409321946445662150301548547825713614803904 100 1750492069977099993617695861204414333904857504132837761