Best Known (39, 39+39, s)-Nets in Base 128
(39, 39+39, 862)-Net over F128 — Constructive and digital
Digital (39, 78, 862)-net over F128, using
- 1281 times duplication [i] based on digital (38, 77, 862)-net over F128, using
- net defined by OOA [i] based on linear OOA(12877, 862, F128, 39, 39) (dual of [(862, 39), 33541, 40]-NRT-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(12877, 16379, F128, 39) (dual of [16379, 16302, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(12877, 16384, F128, 39) (dual of [16384, 16307, 40]-code), using
- an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- discarding factors / shortening the dual code based on linear OA(12877, 16384, F128, 39) (dual of [16384, 16307, 40]-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(12877, 16379, F128, 39) (dual of [16379, 16302, 40]-code), using
- net defined by OOA [i] based on linear OOA(12877, 862, F128, 39, 39) (dual of [(862, 39), 33541, 40]-NRT-code), using
(39, 39+39, 4095)-Net over F128 — Digital
Digital (39, 78, 4095)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12878, 4095, F128, 4, 39) (dual of [(4095, 4), 16302, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12878, 4097, F128, 4, 39) (dual of [(4097, 4), 16310, 40]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12878, 16388, F128, 39) (dual of [16388, 16310, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(12878, 16390, F128, 39) (dual of [16390, 16312, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- linear OA(12877, 16385, F128, 39) (dual of [16385, 16308, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(12873, 16385, F128, 37) (dual of [16385, 16312, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (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([0,19]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12878, 16390, F128, 39) (dual of [16390, 16312, 40]-code), using
- OOA 4-folding [i] based on linear OA(12878, 16388, F128, 39) (dual of [16388, 16310, 40]-code), using
- discarding factors / shortening the dual code based on linear OOA(12878, 4097, F128, 4, 39) (dual of [(4097, 4), 16310, 40]-NRT-code), using
(39, 39+39, large)-Net in Base 128 — Upper bound on s
There is no (39, 78, large)-net in base 128, because
- 37 times m-reduction [i] would yield (39, 41, large)-net in base 128, but