Best Known (69−22, 69, s)-Nets in Base 128
(69−22, 69, 190652)-Net over F128 — Constructive and digital
Digital (47, 69, 190652)-net over F128, using
- net defined by OOA [i] based on linear OOA(12869, 190652, F128, 22, 22) (dual of [(190652, 22), 4194275, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(12869, 2097172, F128, 22) (dual of [2097172, 2097103, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(12869, 2097175, F128, 22) (dual of [2097175, 2097106, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(12864, 2097152, F128, 22) (dual of [2097152, 2097088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(1285, 23, F128, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(12869, 2097175, F128, 22) (dual of [2097175, 2097106, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(12869, 2097172, F128, 22) (dual of [2097172, 2097103, 23]-code), using
(69−22, 69, 1048587)-Net over F128 — Digital
Digital (47, 69, 1048587)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12869, 1048587, F128, 2, 22) (dual of [(1048587, 2), 2097105, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12869, 2097174, F128, 22) (dual of [2097174, 2097105, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(12869, 2097175, F128, 22) (dual of [2097175, 2097106, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(12864, 2097152, F128, 22) (dual of [2097152, 2097088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(1285, 23, F128, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(12869, 2097175, F128, 22) (dual of [2097175, 2097106, 23]-code), using
- OOA 2-folding [i] based on linear OA(12869, 2097174, F128, 22) (dual of [2097174, 2097105, 23]-code), using
(69−22, 69, large)-Net in Base 128 — Upper bound on s
There is no (47, 69, large)-net in base 128, because
- 20 times m-reduction [i] would yield (47, 49, large)-net in base 128, but