Best Known (52−18, 52, s)-Nets in Base 128
(52−18, 52, 233017)-Net over F128 — Constructive and digital
Digital (34, 52, 233017)-net over F128, using
- net defined by OOA [i] based on linear OOA(12852, 233017, F128, 18, 18) (dual of [(233017, 18), 4194254, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(12852, 2097153, F128, 18) (dual of [2097153, 2097101, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12852, 2097155, F128, 18) (dual of [2097155, 2097103, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 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(12849, 2097152, F128, 17) (dual of [2097152, 2097103, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(12852, 2097155, F128, 18) (dual of [2097155, 2097103, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(12852, 2097153, F128, 18) (dual of [2097153, 2097101, 19]-code), using
(52−18, 52, 699051)-Net over F128 — Digital
Digital (34, 52, 699051)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12852, 699051, F128, 3, 18) (dual of [(699051, 3), 2097101, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12852, 2097153, F128, 18) (dual of [2097153, 2097101, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12852, 2097155, F128, 18) (dual of [2097155, 2097103, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 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(12849, 2097152, F128, 17) (dual of [2097152, 2097103, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(12852, 2097155, F128, 18) (dual of [2097155, 2097103, 19]-code), using
- OOA 3-folding [i] based on linear OA(12852, 2097153, F128, 18) (dual of [2097153, 2097101, 19]-code), using
(52−18, 52, large)-Net in Base 128 — Upper bound on s
There is no (34, 52, large)-net in base 128, because
- 16 times m-reduction [i] would yield (34, 36, large)-net in base 128, but