Best Known (65, 65+19, s)-Nets in Base 32
(65, 65+19, 116553)-Net over F32 — Constructive and digital
Digital (65, 84, 116553)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- digital (55, 74, 116509)-net over F32, using
- net defined by OOA [i] based on linear OOA(3274, 116509, F32, 19, 19) (dual of [(116509, 19), 2213597, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3274, 1048582, F32, 19) (dual of [1048582, 1048508, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3274, 1048586, F32, 19) (dual of [1048586, 1048512, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- 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(3265, 1048577, F32, 17) (dual of [1048577, 1048512, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3274, 1048586, F32, 19) (dual of [1048586, 1048512, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3274, 1048582, F32, 19) (dual of [1048582, 1048508, 20]-code), using
- net defined by OOA [i] based on linear OOA(3274, 116509, F32, 19, 19) (dual of [(116509, 19), 2213597, 20]-NRT-code), using
- digital (1, 10, 44)-net over F32, using
(65, 65+19, 233019)-Net in Base 32 — Constructive
(65, 84, 233019)-net in base 32, using
- base change [i] based on digital (41, 60, 233019)-net over F128, using
- net defined by OOA [i] based on linear OOA(12860, 233019, F128, 19, 19) (dual of [(233019, 19), 4427301, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12860, 2097172, F128, 19) (dual of [2097172, 2097112, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12860, 2097176, F128, 19) (dual of [2097176, 2097116, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(12855, 2097153, F128, 19) (dual of [2097153, 2097098, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1285, 23, F128, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12860, 2097176, F128, 19) (dual of [2097176, 2097116, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12860, 2097172, F128, 19) (dual of [2097172, 2097112, 20]-code), using
- net defined by OOA [i] based on linear OOA(12860, 233019, F128, 19, 19) (dual of [(233019, 19), 4427301, 20]-NRT-code), using
(65, 65+19, 2575153)-Net over F32 — Digital
Digital (65, 84, 2575153)-net over F32, using
(65, 65+19, large)-Net in Base 32 — Upper bound on s
There is no (65, 84, large)-net in base 32, because
- 17 times m-reduction [i] would yield (65, 67, large)-net in base 32, but