Best Known (31, 31+11, s)-Nets in Base 32
(31, 31+11, 209717)-Net over F32 — Constructive and digital
Digital (31, 42, 209717)-net over F32, using
- net defined by OOA [i] based on linear OOA(3242, 209717, F32, 11, 11) (dual of [(209717, 11), 2306845, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3242, 1048586, F32, 11) (dual of [1048586, 1048544, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(3241, 1048577, F32, 11) (dual of [1048577, 1048536, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- 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(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([0,5]) ⊂ C([0,4]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(3242, 1048586, F32, 11) (dual of [1048586, 1048544, 12]-code), using
(31, 31+11, 962016)-Net over F32 — Digital
Digital (31, 42, 962016)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3242, 962016, F32, 11) (dual of [962016, 961974, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3242, 1048586, F32, 11) (dual of [1048586, 1048544, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(3241, 1048577, F32, 11) (dual of [1048577, 1048536, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- 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(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([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3242, 1048586, F32, 11) (dual of [1048586, 1048544, 12]-code), using
(31, 31+11, large)-Net in Base 32 — Upper bound on s
There is no (31, 42, large)-net in base 32, because
- 9 times m-reduction [i] would yield (31, 33, large)-net in base 32, but