Best Known (42, 42+21, s)-Nets in Base 128
(42, 42+21, 209716)-Net over F128 — Constructive and digital
Digital (42, 63, 209716)-net over F128, using
- net defined by OOA [i] based on linear OOA(12863, 209716, F128, 21, 21) (dual of [(209716, 21), 4403973, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12863, 2097161, F128, 21) (dual of [2097161, 2097098, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12863, 2097163, F128, 21) (dual of [2097163, 2097100, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] 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]
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(12863, 2097163, F128, 21) (dual of [2097163, 2097100, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12863, 2097161, F128, 21) (dual of [2097161, 2097098, 22]-code), using
(42, 42+21, 823043)-Net over F128 — Digital
Digital (42, 63, 823043)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12863, 823043, F128, 2, 21) (dual of [(823043, 2), 1646023, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12863, 1048581, F128, 2, 21) (dual of [(1048581, 2), 2097099, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12863, 2097162, F128, 21) (dual of [2097162, 2097099, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12863, 2097163, F128, 21) (dual of [2097163, 2097100, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] 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]
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(12863, 2097163, F128, 21) (dual of [2097163, 2097100, 22]-code), using
- OOA 2-folding [i] based on linear OA(12863, 2097162, F128, 21) (dual of [2097162, 2097099, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(12863, 1048581, F128, 2, 21) (dual of [(1048581, 2), 2097099, 22]-NRT-code), using
(42, 42+21, large)-Net in Base 128 — Upper bound on s
There is no (42, 63, large)-net in base 128, because
- 19 times m-reduction [i] would yield (42, 44, large)-net in base 128, but