Best Known (56−16, 56, s)-Nets in Base 128
(56−16, 56, 262294)-Net over F128 — Constructive and digital
Digital (40, 56, 262294)-net over F128, using
- 1281 times duplication [i] based on digital (39, 55, 262294)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- 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
- digital (1, 9, 150)-net over F128, using
- (u, u+v)-construction [i] based on
(56−16, 56, 1048575)-Net in Base 128 — Constructive
(40, 56, 1048575)-net in base 128, using
- t-expansion [i] based on (39, 56, 1048575)-net in base 128, using
- base change [i] based on digital (32, 49, 1048575)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 1048575, F256, 17, 17) (dual of [(1048575, 17), 17825726, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25649, 8388601, F256, 17) (dual of [8388601, 8388552, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25649, 8388601, F256, 17) (dual of [8388601, 8388552, 18]-code), using
- net defined by OOA [i] based on linear OOA(25649, 1048575, F256, 17, 17) (dual of [(1048575, 17), 17825726, 18]-NRT-code), using
- base change [i] based on digital (32, 49, 1048575)-net over F256, using
(56−16, 56, 3722897)-Net over F128 — Digital
Digital (40, 56, 3722897)-net over F128, using
(56−16, 56, 4282917)-Net in Base 128
(40, 56, 4282917)-net in base 128, using
- base change [i] based on digital (33, 49, 4282917)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25649, 4282917, F256, 16) (dual of [4282917, 4282868, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 16) (dual of [large, large−49, 17]-code), using
- strength reduction [i] based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- strength reduction [i] based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 16) (dual of [large, large−49, 17]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25649, 4282917, F256, 16) (dual of [4282917, 4282868, 17]-code), using
(56−16, 56, large)-Net in Base 128 — Upper bound on s
There is no (40, 56, large)-net in base 128, because
- 14 times m-reduction [i] would yield (40, 42, large)-net in base 128, but