Best Known (16, 16+9, s)-Nets in Base 128
(16, 16+9, 524288)-Net over F128 — Constructive and digital
Digital (16, 25, 524288)-net over F128, using
- net defined by OOA [i] based on linear OOA(12825, 524288, F128, 9, 9) (dual of [(524288, 9), 4718567, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using
(16, 16+9, 1048577)-Net over F128 — Digital
Digital (16, 25, 1048577)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12825, 1048577, F128, 2, 9) (dual of [(1048577, 2), 2097129, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12825, 2097154, F128, 9) (dual of [2097154, 2097129, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(12825, 2097155, F128, 9) (dual of [2097155, 2097130, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(12825, 2097152, F128, 9) (dual of [2097152, 2097127, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(12825, 2097155, F128, 9) (dual of [2097155, 2097130, 10]-code), using
- OOA 2-folding [i] based on linear OA(12825, 2097154, F128, 9) (dual of [2097154, 2097129, 10]-code), using
(16, 16+9, large)-Net in Base 128 — Upper bound on s
There is no (16, 25, large)-net in base 128, because
- 7 times m-reduction [i] would yield (16, 18, large)-net in base 128, but