Best Known (42, 42+16, s)-Nets in Base 128
(42, 42+16, 262402)-Net over F128 — Constructive and digital
Digital (42, 58, 262402)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 258)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- digital (0, 8, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 4, 129)-net over F128, using
- (u, u+v)-construction [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
- digital (4, 12, 258)-net over F128, using
(42, 42+16, 1048575)-Net in Base 128 — Constructive
(42, 58, 1048575)-net in base 128, using
- 1282 times duplication [i] based on (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
- t-expansion [i] based on (39, 56, 1048575)-net in base 128, using
(42, 42+16, 7109547)-Net over F128 — Digital
Digital (42, 58, 7109547)-net over F128, using
(42, 42+16, large)-Net in Base 128 — Upper bound on s
There is no (42, 58, large)-net in base 128, because
- 14 times m-reduction [i] would yield (42, 44, large)-net in base 128, but