Best Known (80, 80+27, s)-Nets in Base 32
(80, 80+27, 80660)-Net over F32 — Constructive and digital
Digital (80, 107, 80660)-net over F32, using
- 321 times duplication [i] based on digital (79, 106, 80660)-net over F32, using
- net defined by OOA [i] based on linear OOA(32106, 80660, F32, 27, 27) (dual of [(80660, 27), 2177714, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(32106, 1048581, F32, 27) (dual of [1048581, 1048475, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(32106, 1048586, F32, 27) (dual of [1048586, 1048480, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(32105, 1048577, F32, 27) (dual of [1048577, 1048472, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3297, 1048577, F32, 25) (dual of [1048577, 1048480, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(32106, 1048586, F32, 27) (dual of [1048586, 1048480, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(32106, 1048581, F32, 27) (dual of [1048581, 1048475, 28]-code), using
- net defined by OOA [i] based on linear OOA(32106, 80660, F32, 27, 27) (dual of [(80660, 27), 2177714, 28]-NRT-code), using
(80, 80+27, 790848)-Net over F32 — Digital
Digital (80, 107, 790848)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32107, 790848, F32, 27) (dual of [790848, 790741, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(32107, 1048590, F32, 27) (dual of [1048590, 1048483, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- linear OA(32105, 1048576, F32, 27) (dual of [1048576, 1048471, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(3293, 1048576, F32, 24) (dual of [1048576, 1048483, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(322, 14, F32, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(32107, 1048590, F32, 27) (dual of [1048590, 1048483, 28]-code), using
(80, 80+27, large)-Net in Base 32 — Upper bound on s
There is no (80, 107, large)-net in base 32, because
- 25 times m-reduction [i] would yield (80, 82, large)-net in base 32, but