Best Known (65−21, 65, s)-Nets in Base 128
(65−21, 65, 209717)-Net over F128 — Constructive and digital
Digital (44, 65, 209717)-net over F128, using
- net defined by OOA [i] based on linear OOA(12865, 209717, F128, 21, 21) (dual of [(209717, 21), 4403992, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12865, 2097171, F128, 21) (dual of [2097171, 2097106, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [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(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(20) ⊂ Ce(15) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(12865, 2097171, F128, 21) (dual of [2097171, 2097106, 22]-code), using
(65−21, 65, 1048585)-Net over F128 — Digital
Digital (44, 65, 1048585)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12865, 1048585, F128, 2, 21) (dual of [(1048585, 2), 2097105, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12865, 2097170, F128, 21) (dual of [2097170, 2097105, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12865, 2097171, F128, 21) (dual of [2097171, 2097106, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [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(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(20) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(12865, 2097171, F128, 21) (dual of [2097171, 2097106, 22]-code), using
- OOA 2-folding [i] based on linear OA(12865, 2097170, F128, 21) (dual of [2097170, 2097105, 22]-code), using
(65−21, 65, large)-Net in Base 128 — Upper bound on s
There is no (44, 65, large)-net in base 128, because
- 19 times m-reduction [i] would yield (44, 46, large)-net in base 128, but