Best Known (17, 30, s)-Nets in Base 64
(17, 30, 685)-Net over F64 — Constructive and digital
Digital (17, 30, 685)-net over F64, using
- net defined by OOA [i] based on linear OOA(6430, 685, F64, 13, 13) (dual of [(685, 13), 8875, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6430, 4111, F64, 13) (dual of [4111, 4081, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6430, 4114, F64, 13) (dual of [4114, 4084, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- linear OA(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(6413, 4097, F64, 7) (dual of [4097, 4084, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6430, 4114, F64, 13) (dual of [4114, 4084, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6430, 4111, F64, 13) (dual of [4111, 4081, 14]-code), using
(17, 30, 2730)-Net in Base 64 — Constructive
(17, 30, 2730)-net in base 64, using
- net defined by OOA [i] based on OOA(6430, 2730, S64, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(6430, 16381, S64, 13), using
- discarding factors based on OA(6430, 16386, S64, 13), using
- discarding parts of the base [i] based on linear OA(12825, 16386, F128, 13) (dual of [16386, 16361, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(12825, 16386, F128, 13) (dual of [16386, 16361, 14]-code), using
- discarding factors based on OA(6430, 16386, S64, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(6430, 16381, S64, 13), using
(17, 30, 4233)-Net over F64 — Digital
Digital (17, 30, 4233)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6430, 4233, F64, 13) (dual of [4233, 4203, 14]-code), using
- 130 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 23 times 0, 1, 99 times 0) [i] based on linear OA(6425, 4098, F64, 13) (dual of [4098, 4073, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(6425, 4096, F64, 13) (dual of [4096, 4071, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- 130 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 23 times 0, 1, 99 times 0) [i] based on linear OA(6425, 4098, F64, 13) (dual of [4098, 4073, 14]-code), using
(17, 30, large)-Net in Base 64 — Upper bound on s
There is no (17, 30, large)-net in base 64, because
- 11 times m-reduction [i] would yield (17, 19, large)-net in base 64, but