Best Known (42, 55, s)-Nets in Base 128
(42, 55, 1398229)-Net over F128 — Constructive and digital
Digital (42, 55, 1398229)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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 (36, 49, 1398100)-net over F128, using
- net defined by OOA [i] based on linear OOA(12849, 1398100, F128, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12849, 8388601, F128, 13) (dual of [8388601, 8388552, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12849, large, F128, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(12849, large, F128, 13) (dual of [large, large−49, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12849, 8388601, F128, 13) (dual of [8388601, 8388552, 14]-code), using
- net defined by OOA [i] based on linear OOA(12849, 1398100, F128, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- digital (0, 6, 129)-net over F128, using
(42, 55, 1409023)-Net in Base 128 — Constructive
(42, 55, 1409023)-net in base 128, using
- net defined by OOA [i] based on OOA(12855, 1409023, S128, 14, 13), using
- OOA 3-folding and stacking with additional row [i] based on OOA(12855, 4227070, S128, 2, 13), using
- discarding parts of the base [i] based on linear OOA(25648, 4227070, F256, 2, 13) (dual of [(4227070, 2), 8454092, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(25611, 32769, F256, 2, 6) (dual of [(32769, 2), 65527, 7]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2569, 65536, F256, 5) (dual of [65536, 65527, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- OOA 2-folding [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- linear OOA(25637, 4194301, F256, 2, 13) (dual of [(4194301, 2), 8388565, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25637, 8388602, F256, 13) (dual of [8388602, 8388565, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- OOA 2-folding [i] based on linear OA(25637, 8388602, F256, 13) (dual of [8388602, 8388565, 14]-code), using
- linear OOA(25611, 32769, F256, 2, 6) (dual of [(32769, 2), 65527, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding parts of the base [i] based on linear OOA(25648, 4227070, F256, 2, 13) (dual of [(4227070, 2), 8454092, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on OOA(12855, 4227070, S128, 2, 13), using
(42, 55, large)-Net over F128 — Digital
Digital (42, 55, large)-net over F128, using
- t-expansion [i] based on digital (40, 55, large)-net over F128, using
(42, 55, large)-Net in Base 128 — Upper bound on s
There is no (42, 55, large)-net in base 128, because
- 11 times m-reduction [i] would yield (42, 44, large)-net in base 128, but