Best Known (69−34, 69, s)-Nets in Base 128
(69−34, 69, 964)-Net over F128 — Constructive and digital
Digital (35, 69, 964)-net over F128, using
- 1 times m-reduction [i] based on digital (35, 70, 964)-net over F128, using
- net defined by OOA [i] based on linear OOA(12870, 964, F128, 35, 35) (dual of [(964, 35), 33670, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(12870, 16389, F128, 35) (dual of [16389, 16319, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(12870, 16390, F128, 35) (dual of [16390, 16320, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,16]) [i] based on
- linear OA(12869, 16385, F128, 35) (dual of [16385, 16316, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(12865, 16385, F128, 33) (dual of [16385, 16320, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,17]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12870, 16390, F128, 35) (dual of [16390, 16320, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(12870, 16389, F128, 35) (dual of [16389, 16319, 36]-code), using
- net defined by OOA [i] based on linear OOA(12870, 964, F128, 35, 35) (dual of [(964, 35), 33670, 36]-NRT-code), using
(69−34, 69, 4098)-Net over F128 — Digital
Digital (35, 69, 4098)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12869, 4098, F128, 4, 34) (dual of [(4098, 4), 16323, 35]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12869, 16392, F128, 34) (dual of [16392, 16323, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- linear OA(12867, 16384, F128, 34) (dual of [16384, 16317, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(33) ⊂ Ce(30) [i] based on
- OOA 4-folding [i] based on linear OA(12869, 16392, F128, 34) (dual of [16392, 16323, 35]-code), using
(69−34, 69, large)-Net in Base 128 — Upper bound on s
There is no (35, 69, large)-net in base 128, because
- 32 times m-reduction [i] would yield (35, 37, large)-net in base 128, but