Best Known (42−20, 42, s)-Nets in Base 128
(42−20, 42, 1639)-Net over F128 — Constructive and digital
Digital (22, 42, 1639)-net over F128, using
- 1 times m-reduction [i] based on digital (22, 43, 1639)-net over F128, using
- net defined by OOA [i] based on linear OOA(12843, 1639, F128, 21, 21) (dual of [(1639, 21), 34376, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12843, 16391, F128, 21) (dual of [16391, 16348, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12843, 16392, F128, 21) (dual of [16392, 16349, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-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(12843, 16392, F128, 21) (dual of [16392, 16349, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12843, 16391, F128, 21) (dual of [16391, 16348, 22]-code), using
- net defined by OOA [i] based on linear OOA(12843, 1639, F128, 21, 21) (dual of [(1639, 21), 34376, 22]-NRT-code), using
(42−20, 42, 5465)-Net over F128 — Digital
Digital (22, 42, 5465)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12842, 5465, F128, 3, 20) (dual of [(5465, 3), 16353, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12842, 16395, F128, 20) (dual of [16395, 16353, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(12831, 16384, F128, 16) (dual of [16384, 16353, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- OOA 3-folding [i] based on linear OA(12842, 16395, F128, 20) (dual of [16395, 16353, 21]-code), using
(42−20, 42, large)-Net in Base 128 — Upper bound on s
There is no (22, 42, large)-net in base 128, because
- 18 times m-reduction [i] would yield (22, 24, large)-net in base 128, but