Best Known (55, 55+16, s)-Nets in Base 128
(55, 55+16, 1048725)-Net over F128 — Constructive and digital
Digital (55, 71, 1048725)-net over F128, using
- 1281 times duplication [i] based on digital (54, 70, 1048725)-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 (45, 61, 1048575)-net over F128, using
- net defined by OOA [i] based on linear OOA(12861, 1048575, F128, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(12861, 8388600, F128, 16) (dual of [8388600, 8388539, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(12861, large, F128, 16) (dual of [large, large−61, 17]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9256395 | 1284−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(12861, large, F128, 16) (dual of [large, large−61, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(12861, 8388600, F128, 16) (dual of [8388600, 8388539, 17]-code), using
- net defined by OOA [i] based on linear OOA(12861, 1048575, F128, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- digital (1, 9, 150)-net over F128, using
- (u, u+v)-construction [i] based on
(55, 55+16, 1064959)-Net in Base 128 — Constructive
(55, 71, 1064959)-net in base 128, using
- (u, u+v)-construction [i] based on
- (10, 18, 16384)-net in base 128, using
- net defined by OOA [i] based on OOA(12818, 16384, S128, 8, 8), using
- OA 4-folding and stacking [i] based on OA(12818, 65536, S128, 8), using
- discarding factors based on OA(12818, 65538, S128, 8), using
- discarding parts of the base [i] based on linear OA(25615, 65538, F256, 8) (dual of [65538, 65523, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(25613, 65536, F256, 7) (dual of [65536, 65523, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding parts of the base [i] based on linear OA(25615, 65538, F256, 8) (dual of [65538, 65523, 9]-code), using
- discarding factors based on OA(12818, 65538, S128, 8), using
- OA 4-folding and stacking [i] based on OA(12818, 65536, S128, 8), using
- net defined by OOA [i] based on OOA(12818, 16384, S128, 8, 8), using
- (37, 53, 1048575)-net in base 128, using
- net defined by OOA [i] based on OOA(12853, 1048575, S128, 16, 16), using
- OA 8-folding and stacking [i] based on OA(12853, 8388600, S128, 16), using
- discarding factors based on OA(12853, large, S128, 16), using
- discarding parts of the base [i] based on linear OA(25646, large, F256, 16) (dual of [large, large−46, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding parts of the base [i] based on linear OA(25646, large, F256, 16) (dual of [large, large−46, 17]-code), using
- discarding factors based on OA(12853, large, S128, 16), using
- OA 8-folding and stacking [i] based on OA(12853, 8388600, S128, 16), using
- net defined by OOA [i] based on OOA(12853, 1048575, S128, 16, 16), using
- (10, 18, 16384)-net in base 128, using
(55, 55+16, large)-Net over F128 — Digital
Digital (55, 71, large)-net over F128, using
- t-expansion [i] based on digital (54, 71, large)-net over F128, using
- 3 times m-reduction [i] based on digital (54, 74, large)-net over F128, using
(55, 55+16, large)-Net in Base 128 — Upper bound on s
There is no (55, 71, large)-net in base 128, because
- 14 times m-reduction [i] would yield (55, 57, large)-net in base 128, but