Best Known (45−21, 45, s)-Nets in Base 128
(45−21, 45, 1639)-Net over F128 — Constructive and digital
Digital (24, 45, 1639)-net over F128, using
- 1282 times duplication [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
(45−21, 45, 6422)-Net over F128 — Digital
Digital (24, 45, 6422)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12845, 6422, F128, 2, 21) (dual of [(6422, 2), 12799, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12845, 8199, F128, 2, 21) (dual of [(8199, 2), 16353, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12845, 16398, F128, 21) (dual of [16398, 16353, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [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(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(1284, 14, F128, 4) (dual of [14, 10, 5]-code or 14-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(20) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(12845, 16398, F128, 21) (dual of [16398, 16353, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(12845, 8199, F128, 2, 21) (dual of [(8199, 2), 16353, 22]-NRT-code), using
(45−21, 45, large)-Net in Base 128 — Upper bound on s
There is no (24, 45, large)-net in base 128, because
- 19 times m-reduction [i] would yield (24, 26, large)-net in base 128, but