Best Known (8, 13, s)-Nets in Base 4
(8, 13, 127)-Net over F4 — Constructive and digital
Digital (8, 13, 127)-net over F4, using
- net defined by OOA [i] based on linear OOA(413, 127, F4, 5, 5) (dual of [(127, 5), 622, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(413, 255, F4, 5) (dual of [255, 242, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(413, 255, F4, 5) (dual of [255, 242, 6]-code), using
(8, 13, 154)-Net over F4 — Digital
Digital (8, 13, 154)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(413, 154, F4, 5) (dual of [154, 141, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 255, F4, 5) (dual of [255, 242, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(413, 255, F4, 5) (dual of [255, 242, 6]-code), using
(8, 13, 1929)-Net in Base 4 — Upper bound on s
There is no (8, 13, 1930)-net in base 4, because
- 1 times m-reduction [i] would yield (8, 12, 1930)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 16 782316 > 412 [i]