Best Known (12−4, 12, s)-Nets in Base 32
(12−4, 12, 16432)-Net over F32 — Constructive and digital
Digital (8, 12, 16432)-net over F32, using
- net defined by OOA [i] based on linear OOA(3212, 16432, F32, 4, 4) (dual of [(16432, 4), 65716, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(3212, 32864, F32, 4) (dual of [32864, 32852, 5]-code), using
- generalized (u, u+v)-construction [i] based on
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(321, 1027, F32, 1) (dual of [1027, 1026, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(321, 1027, F32, 1) (dual of [1027, 1026, 2]-code) (see above)
- linear OA(323, 1027, F32, 2) (dual of [1027, 1024, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(323, 1057, F32, 2) (dual of [1057, 1054, 3]-code), using
- Hamming code H(3,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 1057, F32, 2) (dual of [1057, 1054, 3]-code), using
- linear OA(327, 1027, F32, 4) (dual of [1027, 1020, 5]-code), using
- construction XX applied to C1 = C([1022,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([1022,2]) [i] based on
- linear OA(325, 1023, F32, 3) (dual of [1023, 1018, 4]-code or 1023-cap in PG(4,32)), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(325, 1023, F32, 3) (dual of [1023, 1018, 4]-code or 1023-cap in PG(4,32)), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(327, 1023, F32, 4) (dual of [1023, 1016, 5]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(323, 1023, F32, 2) (dual of [1023, 1020, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s (see above)
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([1022,2]) [i] based on
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code), using
- generalized (u, u+v)-construction [i] based on
- OA 2-folding and stacking [i] based on linear OA(3212, 32864, F32, 4) (dual of [32864, 32852, 5]-code), using
(12−4, 12, 61466)-Net over F32 — Digital
Digital (8, 12, 61466)-net over F32, using
(12−4, 12, 131073)-Net in Base 32 — Constructive
(8, 12, 131073)-net in base 32, using
- base change [i] based on digital (6, 10, 131073)-net over F64, using
- net defined by OOA [i] based on linear OOA(6410, 131073, F64, 4, 4) (dual of [(131073, 4), 524282, 5]-NRT-code), using
- appending kth column [i] based on 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
- appending kth column [i] based on linear OOA(6410, 131073, F64, 3, 4) (dual of [(131073, 3), 393209, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6410, 131073, F64, 4, 4) (dual of [(131073, 4), 524282, 5]-NRT-code), using
(12−4, 12, 262147)-Net in Base 32
(8, 12, 262147)-net in base 32, using
- base change [i] based on digital (6, 10, 262147)-net over F64, using
- net defined by OOA [i] based on linear OOA(6410, 262147, F64, 4, 4) (dual of [(262147, 4), 1048578, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(6410, 262147, F64, 3, 4) (dual of [(262147, 3), 786431, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] 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
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6410, 262147, F64, 4) (dual of [262147, 262137, 5]-code), using
- appending kth column [i] based on linear OOA(6410, 262147, F64, 3, 4) (dual of [(262147, 3), 786431, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6410, 262147, F64, 4, 4) (dual of [(262147, 4), 1048578, 5]-NRT-code), using
(12−4, 12, large)-Net in Base 32 — Upper bound on s
There is no (8, 12, large)-net in base 32, because
- 2 times m-reduction [i] would yield (8, 10, large)-net in base 32, but