Best Known (13−7, 13, s)-Nets in Base 128
(13−7, 13, 5461)-Net over F128 — Constructive and digital
Digital (6, 13, 5461)-net over F128, using
- net defined by OOA [i] based on linear OOA(12813, 5461, F128, 7, 7) (dual of [(5461, 7), 38214, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using
(13−7, 13, 8193)-Net over F128 — Digital
Digital (6, 13, 8193)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12813, 8193, F128, 2, 7) (dual of [(8193, 2), 16373, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12813, 16386, F128, 7) (dual of [16386, 16373, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(12811, 16384, F128, 6) (dual of [16384, 16373, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 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(6) ⊂ Ce(5) [i] based on
- OOA 2-folding [i] based on linear OA(12813, 16386, F128, 7) (dual of [16386, 16373, 8]-code), using
(13−7, 13, 3840783)-Net in Base 128 — Upper bound on s
There is no (6, 13, 3840784)-net in base 128, because
- 1 times m-reduction [i] would yield (6, 12, 3840784)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 19 342825 373850 917118 817465 > 12812 [i]