Best Known (64−32, 64, s)-Nets in Base 128
(64−32, 64, 1024)-Net over F128 — Constructive and digital
Digital (32, 64, 1024)-net over F128, using
- 1 times m-reduction [i] based on digital (32, 65, 1024)-net over F128, using
- net defined by OOA [i] based on linear OOA(12865, 1024, F128, 33, 33) (dual of [(1024, 33), 33727, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(12865, 16385, F128, 33) (dual of [16385, 16320, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(12865, 16385, F128, 33) (dual of [16385, 16320, 34]-code), using
- net defined by OOA [i] based on linear OOA(12865, 1024, F128, 33, 33) (dual of [(1024, 33), 33727, 34]-NRT-code), using
(64−32, 64, 4097)-Net over F128 — Digital
Digital (32, 64, 4097)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12864, 4097, F128, 4, 32) (dual of [(4097, 4), 16324, 33]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12864, 16388, F128, 32) (dual of [16388, 16324, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(12864, 16389, F128, 32) (dual of [16389, 16325, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(29) [i] based on
- linear OA(12863, 16384, F128, 32) (dual of [16384, 16321, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(12859, 16384, F128, 30) (dual of [16384, 16325, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 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 Ce(31) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(12864, 16389, F128, 32) (dual of [16389, 16325, 33]-code), using
- OOA 4-folding [i] based on linear OA(12864, 16388, F128, 32) (dual of [16388, 16324, 33]-code), using
(64−32, 64, large)-Net in Base 128 — Upper bound on s
There is no (32, 64, large)-net in base 128, because
- 30 times m-reduction [i] would yield (32, 34, large)-net in base 128, but