Best Known (39, 39+20, s)-Nets in Base 128
(39, 39+20, 209715)-Net over F128 — Constructive and digital
Digital (39, 59, 209715)-net over F128, using
- 1281 times duplication [i] based on digital (38, 58, 209715)-net over F128, using
- net defined by OOA [i] based on linear OOA(12858, 209715, F128, 20, 20) (dual of [(209715, 20), 4194242, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(12858, 2097150, F128, 20) (dual of [2097150, 2097092, 21]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(12858, 2097150, F128, 20) (dual of [2097150, 2097092, 21]-code), using
- net defined by OOA [i] based on linear OOA(12858, 209715, F128, 20, 20) (dual of [(209715, 20), 4194242, 21]-NRT-code), using
(39, 39+20, 699053)-Net over F128 — Digital
Digital (39, 59, 699053)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12859, 699053, F128, 3, 20) (dual of [(699053, 3), 2097100, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12859, 2097159, F128, 20) (dual of [2097159, 2097100, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [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(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 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 Ce(19) ⊂ Ce(17) [i] based on
- OOA 3-folding [i] based on linear OA(12859, 2097159, F128, 20) (dual of [2097159, 2097100, 21]-code), using
(39, 39+20, large)-Net in Base 128 — Upper bound on s
There is no (39, 59, large)-net in base 128, because
- 18 times m-reduction [i] would yield (39, 41, large)-net in base 128, but