Best Known (12−4, 12, s)-Nets in Base 64
(12−4, 12, 131138)-Net over F64 — Constructive and digital
Digital (8, 12, 131138)-net over F64, using
- net defined by OOA [i] based on linear OOA(6412, 131138, F64, 4, 4) (dual of [(131138, 4), 524540, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(6412, 131138, F64, 3, 4) (dual of [(131138, 3), 393402, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(642, 65, F64, 3, 2) (dual of [(65, 3), 193, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;193,64) [i]
- linear OOA(6410, 131073, F64, 3, 4) (dual of [(131073, 3), 393209, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(6410, 262146, F64, 4) (dual of [262146, 262136, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(6410, 262147, F64, 4) (dual of [262147, 262137, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(6410, 262144, F64, 4) (dual of [262144, 262134, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(647, 262144, F64, 3) (dual of [262144, 262137, 4]-code or 262144-cap in PG(6,64)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(6410, 262147, F64, 4) (dual of [262147, 262137, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(6410, 262146, F64, 4) (dual of [262146, 262136, 5]-code), using
- linear OOA(642, 65, F64, 3, 2) (dual of [(65, 3), 193, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(6412, 131138, F64, 3, 4) (dual of [(131138, 3), 393402, 5]-NRT-code), using
(12−4, 12, 483910)-Net over F64 — Digital
Digital (8, 12, 483910)-net over F64, using
- net defined by OOA [i] based on linear OOA(6412, 483910, F64, 4, 4) (dual of [(483910, 4), 1935628, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(6412, 483910, F64, 3, 4) (dual of [(483910, 3), 1451718, 5]-NRT-code), using
(12−4, 12, 1048577)-Net in Base 64 — Constructive
(8, 12, 1048577)-net in base 64, using
- net defined by OOA [i] based on OOA(6412, 1048577, S64, 4, 4), using
- appending kth column [i] based on OOA(6412, 1048577, S64, 3, 4), using
- OA 2-folding and stacking [i] based on OA(6412, 2097154, S64, 4), using
- discarding factors based on OA(6412, 2097155, S64, 4), using
- discarding parts of the base [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1287, 2097152, F128, 3) (dual of [2097152, 2097145, 4]-code or 2097152-cap in PG(6,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 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(3) ⊂ Ce(2) [i] based on
- discarding parts of the base [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- discarding factors based on OA(6412, 2097155, S64, 4), using
- OA 2-folding and stacking [i] based on OA(6412, 2097154, S64, 4), using
- appending kth column [i] based on OOA(6412, 1048577, S64, 3, 4), using
(12−4, 12, 2097155)-Net in Base 64
(8, 12, 2097155)-net in base 64, using
- net defined by OOA [i] based on OOA(6412, 2097155, S64, 4, 4), using
- appending kth column [i] based on OOA(6412, 2097155, S64, 3, 4), using
- discarding parts of the base [i] based on linear OOA(12810, 2097155, F128, 3, 4) (dual of [(2097155, 3), 6291455, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1287, 2097152, F128, 3) (dual of [2097152, 2097145, 4]-code or 2097152-cap in PG(6,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 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(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- discarding parts of the base [i] based on linear OOA(12810, 2097155, F128, 3, 4) (dual of [(2097155, 3), 6291455, 5]-NRT-code), using
- appending kth column [i] based on OOA(6412, 2097155, S64, 3, 4), using
(12−4, 12, large)-Net in Base 64 — Upper bound on s
There is no (8, 12, large)-net in base 64, because
- 2 times m-reduction [i] would yield (8, 10, large)-net in base 64, but