Best Known (76−26, 76, s)-Nets in Base 128
(76−26, 76, 161319)-Net over F128 — Constructive and digital
Digital (50, 76, 161319)-net over F128, using
- net defined by OOA [i] based on linear OOA(12876, 161319, F128, 26, 26) (dual of [(161319, 26), 4194218, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(12876, 2097147, F128, 26) (dual of [2097147, 2097071, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(12876, 2097152, F128, 26) (dual of [2097152, 2097076, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(12876, 2097152, F128, 26) (dual of [2097152, 2097076, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(12876, 2097147, F128, 26) (dual of [2097147, 2097071, 27]-code), using
(76−26, 76, 699051)-Net over F128 — Digital
Digital (50, 76, 699051)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12876, 699051, F128, 3, 26) (dual of [(699051, 3), 2097077, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12876, 2097153, F128, 26) (dual of [2097153, 2097077, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(12876, 2097155, F128, 26) (dual of [2097155, 2097079, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(12876, 2097152, F128, 26) (dual of [2097152, 2097076, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(12873, 2097152, F128, 25) (dual of [2097152, 2097079, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(12876, 2097155, F128, 26) (dual of [2097155, 2097079, 27]-code), using
- OOA 3-folding [i] based on linear OA(12876, 2097153, F128, 26) (dual of [2097153, 2097077, 27]-code), using
(76−26, 76, large)-Net in Base 128 — Upper bound on s
There is no (50, 76, large)-net in base 128, because
- 24 times m-reduction [i] would yield (50, 52, large)-net in base 128, but