Best Known (32, 32+13, s)-Nets in Base 8
(32, 32+13, 683)-Net over F8 — Constructive and digital
Digital (32, 45, 683)-net over F8, using
- net defined by OOA [i] based on linear OOA(845, 683, F8, 13, 13) (dual of [(683, 13), 8834, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(845, 4099, F8, 13) (dual of [4099, 4054, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(845, 4100, F8, 13) (dual of [4100, 4055, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(845, 4096, F8, 13) (dual of [4096, 4051, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(841, 4096, F8, 12) (dual of [4096, 4055, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(845, 4100, F8, 13) (dual of [4100, 4055, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(845, 4099, F8, 13) (dual of [4099, 4054, 14]-code), using
(32, 32+13, 2867)-Net over F8 — Digital
Digital (32, 45, 2867)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(845, 2867, F8, 13) (dual of [2867, 2822, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(845, 4096, F8, 13) (dual of [4096, 4051, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(845, 4096, F8, 13) (dual of [4096, 4051, 14]-code), using
(32, 32+13, 1793837)-Net in Base 8 — Upper bound on s
There is no (32, 45, 1793838)-net in base 8, because
- 1 times m-reduction [i] would yield (32, 44, 1793838)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 5444 524275 825522 785562 747474 397053 994324 > 844 [i]