Best Known (37, 37+19, s)-Nets in Base 128
(37, 37+19, 233017)-Net over F128 — Constructive and digital
Digital (37, 56, 233017)-net over F128, using
- 1281 times duplication [i] based on digital (36, 55, 233017)-net over F128, using
- net defined by OOA [i] based on linear OOA(12855, 233017, F128, 19, 19) (dual of [(233017, 19), 4427268, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12855, 2097154, F128, 19) (dual of [2097154, 2097099, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12855, 2097155, F128, 19) (dual of [2097155, 2097100, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(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(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(12855, 2097155, F128, 19) (dual of [2097155, 2097100, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12855, 2097154, F128, 19) (dual of [2097154, 2097099, 20]-code), using
- net defined by OOA [i] based on linear OOA(12855, 233017, F128, 19, 19) (dual of [(233017, 19), 4427268, 20]-NRT-code), using
(37, 37+19, 699053)-Net over F128 — Digital
Digital (37, 56, 699053)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12856, 699053, F128, 3, 19) (dual of [(699053, 3), 2097103, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12856, 2097159, F128, 19) (dual of [2097159, 2097103, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12856, 2097160, F128, 19) (dual of [2097160, 2097104, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(12855, 2097153, F128, 19) (dual of [2097153, 2097098, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [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 C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12856, 2097160, F128, 19) (dual of [2097160, 2097104, 20]-code), using
- OOA 3-folding [i] based on linear OA(12856, 2097159, F128, 19) (dual of [2097159, 2097103, 20]-code), using
(37, 37+19, large)-Net in Base 128 — Upper bound on s
There is no (37, 56, large)-net in base 128, because
- 17 times m-reduction [i] would yield (37, 39, large)-net in base 128, but