Best Known (31, 47, s)-Nets in Base 128
(31, 47, 262144)-Net over F128 — Constructive and digital
Digital (31, 47, 262144)-net over F128, using
- 1281 times duplication [i] based on digital (30, 46, 262144)-net over F128, using
- net defined by OOA [i] based on linear OOA(12846, 262144, F128, 16, 16) (dual of [(262144, 16), 4194258, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OA 8-folding and stacking [i] based on linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using
- net defined by OOA [i] based on linear OOA(12846, 262144, F128, 16, 16) (dual of [(262144, 16), 4194258, 17]-NRT-code), using
(31, 47, 878594)-Net over F128 — Digital
Digital (31, 47, 878594)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12847, 878594, F128, 2, 16) (dual of [(878594, 2), 1757141, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12847, 1048579, F128, 2, 16) (dual of [(1048579, 2), 2097111, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12847, 2097158, F128, 16) (dual of [2097158, 2097111, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(12847, 2097159, F128, 16) (dual of [2097159, 2097112, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(12847, 2097159, F128, 16) (dual of [2097159, 2097112, 17]-code), using
- OOA 2-folding [i] based on linear OA(12847, 2097158, F128, 16) (dual of [2097158, 2097111, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(12847, 1048579, F128, 2, 16) (dual of [(1048579, 2), 2097111, 17]-NRT-code), using
(31, 47, large)-Net in Base 128 — Upper bound on s
There is no (31, 47, large)-net in base 128, because
- 14 times m-reduction [i] would yield (31, 33, large)-net in base 128, but