Best Known (46, 46+20, s)-Nets in Base 128
(46, 46+20, 209718)-Net over F128 — Constructive and digital
Digital (46, 66, 209718)-net over F128, using
- 1281 times duplication [i] based on digital (45, 65, 209718)-net over F128, using
- net defined by OOA [i] based on linear OOA(12865, 209718, F128, 20, 20) (dual of [(209718, 20), 4194295, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(12865, 2097180, F128, 20) (dual of [2097180, 2097115, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(12865, 2097183, F128, 20) (dual of [2097183, 2097118, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(11) [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(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1287, 31, F128, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to Ce(19) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(12865, 2097183, F128, 20) (dual of [2097183, 2097118, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(12865, 2097180, F128, 20) (dual of [2097180, 2097115, 21]-code), using
- net defined by OOA [i] based on linear OOA(12865, 209718, F128, 20, 20) (dual of [(209718, 20), 4194295, 21]-NRT-code), using
(46, 46+20, 2097187)-Net over F128 — Digital
Digital (46, 66, 2097187)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12866, 2097187, F128, 20) (dual of [2097187, 2097121, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(10) [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(12831, 2097152, F128, 11) (dual of [2097152, 2097121, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(1288, 35, F128, 8) (dual of [35, 27, 9]-code or 35-arc in PG(7,128)), using
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- Reed–Solomon code RS(120,128) [i]
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- construction X applied to Ce(19) ⊂ Ce(10) [i] based on
(46, 46+20, large)-Net in Base 128 — Upper bound on s
There is no (46, 66, large)-net in base 128, because
- 18 times m-reduction [i] would yield (46, 48, large)-net in base 128, but