Best Known (55, 55+19, s)-Nets in Base 32
(55, 55+19, 116509)-Net over F32 — Constructive and digital
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
(55, 55+19, 672774)-Net over F32 — Digital
Digital (55, 74, 672774)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3274, 672774, F32, 19) (dual of [672774, 672700, 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
(55, 55+19, large)-Net in Base 32 — Upper bound on s
There is no (55, 74, large)-net in base 32, because
- 17 times m-reduction [i] would yield (55, 57, large)-net in base 32, but