Best Known (74−21, 74, s)-Nets in Base 128
(74−21, 74, 209907)-Net over F128 — Constructive and digital
Digital (53, 74, 209907)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (3, 13, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- digital (40, 61, 209715)-net over F128, using
- net defined by OOA [i] based on linear OOA(12861, 209715, F128, 21, 21) (dual of [(209715, 21), 4403954, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12861, 2097151, F128, 21) (dual of [2097151, 2097090, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12861, 2097151, F128, 21) (dual of [2097151, 2097090, 22]-code), using
- net defined by OOA [i] based on linear OOA(12861, 209715, F128, 21, 21) (dual of [(209715, 21), 4403954, 22]-NRT-code), using
- digital (3, 13, 192)-net over F128, using
(74−21, 74, 838860)-Net in Base 128 — Constructive
(53, 74, 838860)-net in base 128, using
- 1282 times duplication [i] based on (51, 72, 838860)-net in base 128, using
- base change [i] based on digital (42, 63, 838860)-net over F256, using
- 2562 times duplication [i] based on digital (40, 61, 838860)-net over F256, using
- net defined by OOA [i] based on linear OOA(25661, 838860, F256, 21, 21) (dual of [(838860, 21), 17615999, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(25661, 8388601, F256, 21) (dual of [8388601, 8388540, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(25661, 8388601, F256, 21) (dual of [8388601, 8388540, 22]-code), using
- net defined by OOA [i] based on linear OOA(25661, 838860, F256, 21, 21) (dual of [(838860, 21), 17615999, 22]-NRT-code), using
- 2562 times duplication [i] based on digital (40, 61, 838860)-net over F256, using
- base change [i] based on digital (42, 63, 838860)-net over F256, using
(74−21, 74, 4094308)-Net over F128 — Digital
Digital (53, 74, 4094308)-net over F128, using
(74−21, 74, 4194302)-Net in Base 128
(53, 74, 4194302)-net in base 128, using
- 1282 times duplication [i] based on (51, 72, 4194302)-net in base 128, using
- base change [i] based on digital (42, 63, 4194302)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25663, 4194302, F256, 2, 21) (dual of [(4194302, 2), 8388541, 22]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(25661, 4194301, F256, 2, 21) (dual of [(4194301, 2), 8388541, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25661, 8388602, F256, 21) (dual of [8388602, 8388541, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- OOA 2-folding [i] based on linear OA(25661, 8388602, F256, 21) (dual of [8388602, 8388541, 22]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(25661, 4194301, F256, 2, 21) (dual of [(4194301, 2), 8388541, 22]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25663, 4194302, F256, 2, 21) (dual of [(4194302, 2), 8388541, 22]-NRT-code), using
- base change [i] based on digital (42, 63, 4194302)-net over F256, using
(74−21, 74, large)-Net in Base 128 — Upper bound on s
There is no (53, 74, large)-net in base 128, because
- 19 times m-reduction [i] would yield (53, 55, large)-net in base 128, but