Best Known (69, 69+27, s)-Nets in Base 32
(69, 69+27, 2585)-Net over F32 — Constructive and digital
Digital (69, 96, 2585)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (3, 16, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- digital (53, 80, 2521)-net over F32, using
- net defined by OOA [i] based on linear OOA(3280, 2521, F32, 27, 27) (dual of [(2521, 27), 67987, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3280, 32774, F32, 27) (dual of [32774, 32694, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3280, 32776, F32, 27) (dual of [32776, 32696, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(3279, 32769, F32, 27) (dual of [32769, 32690, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3273, 32769, F32, 25) (dual of [32769, 32696, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(321, 7, F32, 1) (dual of [7, 6, 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
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3280, 32776, F32, 27) (dual of [32776, 32696, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3280, 32774, F32, 27) (dual of [32774, 32694, 28]-code), using
- net defined by OOA [i] based on linear OOA(3280, 2521, F32, 27, 27) (dual of [(2521, 27), 67987, 28]-NRT-code), using
- digital (3, 16, 64)-net over F32, using
(69, 69+27, 20165)-Net in Base 32 — Constructive
(69, 96, 20165)-net in base 32, using
- base change [i] based on digital (53, 80, 20165)-net over F64, using
- 641 times duplication [i] based on digital (52, 79, 20165)-net over F64, using
- net defined by OOA [i] based on linear OOA(6479, 20165, F64, 27, 27) (dual of [(20165, 27), 544376, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(6479, 262146, F64, 27) (dual of [262146, 262067, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(6479, 262147, F64, 27) (dual of [262147, 262068, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(6479, 262144, F64, 27) (dual of [262144, 262065, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(6476, 262144, F64, 26) (dual of [262144, 262068, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(6479, 262147, F64, 27) (dual of [262147, 262068, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(6479, 262146, F64, 27) (dual of [262146, 262067, 28]-code), using
- net defined by OOA [i] based on linear OOA(6479, 20165, F64, 27, 27) (dual of [(20165, 27), 544376, 28]-NRT-code), using
- 641 times duplication [i] based on digital (52, 79, 20165)-net over F64, using
(69, 69+27, 122872)-Net over F32 — Digital
Digital (69, 96, 122872)-net over F32, using
(69, 69+27, large)-Net in Base 32 — Upper bound on s
There is no (69, 96, large)-net in base 32, because
- 25 times m-reduction [i] would yield (69, 71, large)-net in base 32, but