Best Known (27, 27+9, s)-Nets in Base 32
(27, 27+9, 262148)-Net over F32 — Constructive and digital
Digital (27, 36, 262148)-net over F32, using
- net defined by OOA [i] based on linear OOA(3236, 262148, F32, 9, 9) (dual of [(262148, 9), 2359296, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3236, 1048593, F32, 9) (dual of [1048593, 1048557, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(3236, 1048596, F32, 9) (dual of [1048596, 1048560, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- linear OA(3233, 1048577, F32, 9) (dual of [1048577, 1048544, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(3217, 1048577, F32, 5) (dual of [1048577, 1048560, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(323, 19, F32, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,32) or 19-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3236, 1048596, F32, 9) (dual of [1048596, 1048560, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3236, 1048593, F32, 9) (dual of [1048593, 1048557, 10]-code), using
(27, 27+9, 524288)-Net in Base 32 — Constructive
(27, 36, 524288)-net in base 32, using
- 321 times duplication [i] based on (26, 35, 524288)-net in base 32, using
- base change [i] based on digital (16, 25, 524288)-net over F128, using
- net defined by OOA [i] based on linear OOA(12825, 524288, F128, 9, 9) (dual of [(524288, 9), 4718567, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using
- net defined by OOA [i] based on linear OOA(12825, 524288, F128, 9, 9) (dual of [(524288, 9), 4718567, 10]-NRT-code), using
- base change [i] based on digital (16, 25, 524288)-net over F128, using
(27, 27+9, 1048596)-Net over F32 — Digital
Digital (27, 36, 1048596)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3236, 1048596, F32, 9) (dual of [1048596, 1048560, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- linear OA(3233, 1048577, F32, 9) (dual of [1048577, 1048544, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(3217, 1048577, F32, 5) (dual of [1048577, 1048560, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(323, 19, F32, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,32) or 19-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
(27, 27+9, large)-Net in Base 32 — Upper bound on s
There is no (27, 36, large)-net in base 32, because
- 7 times m-reduction [i] would yield (27, 29, large)-net in base 32, but