Best Known (7, 7+6, s)-Nets in Base 64
(7, 7+6, 1368)-Net over F64 — Constructive and digital
Digital (7, 13, 1368)-net over F64, using
- net defined by OOA [i] based on linear OOA(6413, 1368, F64, 6, 6) (dual of [(1368, 6), 8195, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(6413, 4104, F64, 6) (dual of [4104, 4091, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(6411, 4096, F64, 6) (dual of [4096, 4085, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(645, 4096, F64, 3) (dual of [4096, 4091, 4]-code or 4096-cap in PG(4,64)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- OA 3-folding and stacking [i] based on linear OA(6413, 4104, F64, 6) (dual of [4104, 4091, 7]-code), using
(7, 7+6, 4159)-Net over F64 — Digital
Digital (7, 13, 4159)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6413, 4159, F64, 6) (dual of [4159, 4146, 7]-code), using
- 59 step Varšamov–Edel lengthening with (ri) = (2, 58 times 0) [i] based on linear OA(6411, 4098, F64, 6) (dual of [4098, 4087, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(6411, 4096, F64, 6) (dual of [4096, 4085, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(649, 4096, F64, 5) (dual of [4096, 4087, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- 59 step Varšamov–Edel lengthening with (ri) = (2, 58 times 0) [i] based on linear OA(6411, 4098, F64, 6) (dual of [4098, 4087, 7]-code), using
(7, 7+6, 5462)-Net in Base 64 — Constructive
(7, 13, 5462)-net in base 64, using
- net defined by OOA [i] based on OOA(6413, 5462, S64, 6, 6), using
- OA 3-folding and stacking [i] based on OA(6413, 16386, S64, 6), using
- discarding parts of the base [i] based on linear OA(12811, 16386, F128, 6) (dual of [16386, 16375, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(12811, 16384, F128, 6) (dual of [16384, 16373, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1289, 16384, F128, 5) (dual of [16384, 16375, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- discarding parts of the base [i] based on linear OA(12811, 16386, F128, 6) (dual of [16386, 16375, 7]-code), using
- OA 3-folding and stacking [i] based on OA(6413, 16386, S64, 6), using
(7, 7+6, 8192)-Net in Base 64
(7, 13, 8192)-net in base 64, using
- net defined by OOA [i] based on OOA(6413, 8192, S64, 9, 6), using
- OOA stacking with additional row [i] based on OOA(6413, 8193, S64, 3, 6), using
- discarding parts of the base [i] based on linear OOA(12811, 8193, F128, 3, 6) (dual of [(8193, 3), 24568, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12811, 8193, F128, 2, 6) (dual of [(8193, 2), 16375, 7]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12811, 16386, F128, 6) (dual of [16386, 16375, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(12811, 16384, F128, 6) (dual of [16384, 16373, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1289, 16384, F128, 5) (dual of [16384, 16375, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- OOA 2-folding [i] based on linear OA(12811, 16386, F128, 6) (dual of [16386, 16375, 7]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12811, 8193, F128, 2, 6) (dual of [(8193, 2), 16375, 7]-NRT-code), using
- discarding parts of the base [i] based on linear OOA(12811, 8193, F128, 3, 6) (dual of [(8193, 3), 24568, 7]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(6413, 8193, S64, 3, 6), using
(7, 7+6, 1935632)-Net in Base 64 — Upper bound on s
There is no (7, 13, 1935633)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 302231 798889 663047 657680 > 6413 [i]