Best Known (78, 78+21, s)-Nets in Base 32
(78, 78+21, 104956)-Net over F32 — Constructive and digital
Digital (78, 99, 104956)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 17, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (61, 82, 104858)-net over F32, using
- net defined by OOA [i] based on linear OOA(3282, 104858, F32, 21, 21) (dual of [(104858, 21), 2201936, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3282, 1048581, F32, 21) (dual of [1048581, 1048499, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3282, 1048586, F32, 21) (dual of [1048586, 1048504, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3281, 1048577, F32, 21) (dual of [1048577, 1048496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3273, 1048577, F32, 19) (dual of [1048577, 1048504, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(321, 9, F32, 1) (dual of [9, 8, 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([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3282, 1048586, F32, 21) (dual of [1048586, 1048504, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3282, 1048581, F32, 21) (dual of [1048581, 1048499, 22]-code), using
- net defined by OOA [i] based on linear OOA(3282, 104858, F32, 21, 21) (dual of [(104858, 21), 2201936, 22]-NRT-code), using
- digital (7, 17, 98)-net over F32, using
(78, 78+21, 838860)-Net in Base 32 — Constructive
(78, 99, 838860)-net in base 32, using
- 321 times duplication [i] based on (77, 98, 838860)-net in base 32, using
- net defined by OOA [i] based on OOA(3298, 838860, S32, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(3298, 8388601, S32, 21), using
- discarding factors based on OA(3298, large, S32, 21), using
- discarding parts of the base [i] based on linear OA(6481, large, F64, 21) (dual of [large, large−81, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- discarding parts of the base [i] based on linear OA(6481, large, F64, 21) (dual of [large, large−81, 22]-code), using
- discarding factors based on OA(3298, large, S32, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(3298, 8388601, S32, 21), using
- net defined by OOA [i] based on OOA(3298, 838860, S32, 21, 21), using
(78, 78+21, 7558533)-Net over F32 — Digital
Digital (78, 99, 7558533)-net over F32, using
(78, 78+21, large)-Net in Base 32 — Upper bound on s
There is no (78, 99, large)-net in base 32, because
- 19 times m-reduction [i] would yield (78, 80, large)-net in base 32, but