Best Known (42, 42+20, s)-Nets in Base 128
(42, 42+20, 209717)-Net over F128 — Constructive and digital
Digital (42, 62, 209717)-net over F128, using
- net defined by OOA [i] based on linear OOA(12862, 209717, F128, 20, 20) (dual of [(209717, 20), 4194278, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(12862, 2097170, F128, 20) (dual of [2097170, 2097108, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(12862, 2097171, F128, 20) (dual of [2097171, 2097109, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(12862, 2097171, F128, 20) (dual of [2097171, 2097109, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(12862, 2097170, F128, 20) (dual of [2097170, 2097108, 21]-code), using
(42, 42+20, 1048585)-Net over F128 — Digital
Digital (42, 62, 1048585)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12862, 1048585, F128, 2, 20) (dual of [(1048585, 2), 2097108, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12862, 2097170, F128, 20) (dual of [2097170, 2097108, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(12862, 2097171, F128, 20) (dual of [2097171, 2097109, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(12862, 2097171, F128, 20) (dual of [2097171, 2097109, 21]-code), using
- OOA 2-folding [i] based on linear OA(12862, 2097170, F128, 20) (dual of [2097170, 2097108, 21]-code), using
(42, 42+20, large)-Net in Base 128 — Upper bound on s
There is no (42, 62, large)-net in base 128, because
- 18 times m-reduction [i] would yield (42, 44, large)-net in base 128, but