Best Known (159−16, 159, s)-Nets in Base 8
(159−16, 159, 2105345)-Net over F8 — Constructive and digital
Digital (143, 159, 2105345)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (29, 37, 8195)-net over F8, using
- net defined by OOA [i] based on linear OOA(837, 8195, F8, 8, 8) (dual of [(8195, 8), 65523, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(837, 32780, F8, 8) (dual of [32780, 32743, 9]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(826, 32768, F8, 6) (dual of [32768, 32742, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(811, 12, F8, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,8)), using
- dual of repetition code with length 12 [i]
- linear OA(81, 12, F8, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(5) [i] based on
- OA 4-folding and stacking [i] based on linear OA(837, 32780, F8, 8) (dual of [32780, 32743, 9]-code), using
- net defined by OOA [i] based on linear OOA(837, 8195, F8, 8, 8) (dual of [(8195, 8), 65523, 9]-NRT-code), using
- digital (106, 122, 2097150)-net over F8, using
- net defined by OOA [i] based on linear OOA(8122, 2097150, F8, 18, 16) (dual of [(2097150, 18), 37748578, 17]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(8122, 8388601, F8, 2, 16) (dual of [(8388601, 2), 16777080, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8122, 8388602, F8, 2, 16) (dual of [(8388602, 2), 16777082, 17]-NRT-code), using
- trace code [i] based on linear OOA(6461, 4194301, F64, 2, 16) (dual of [(4194301, 2), 8388541, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6461, 8388602, F64, 16) (dual of [8388602, 8388541, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- OOA 2-folding [i] based on linear OA(6461, 8388602, F64, 16) (dual of [8388602, 8388541, 17]-code), using
- trace code [i] based on linear OOA(6461, 4194301, F64, 2, 16) (dual of [(4194301, 2), 8388541, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8122, 8388602, F8, 2, 16) (dual of [(8388602, 2), 16777082, 17]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(8122, 8388601, F8, 2, 16) (dual of [(8388601, 2), 16777080, 17]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8122, 2097150, F8, 18, 16) (dual of [(2097150, 18), 37748578, 17]-NRT-code), using
- digital (29, 37, 8195)-net over F8, using
(159−16, 159, large)-Net over F8 — Digital
Digital (143, 159, large)-net over F8, using
- t-expansion [i] based on digital (138, 159, large)-net over F8, using
- 2 times m-reduction [i] based on digital (138, 161, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8161, large, F8, 23) (dual of [large, large−161, 24]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8161, large, F8, 23) (dual of [large, large−161, 24]-code), using
- 2 times m-reduction [i] based on digital (138, 161, large)-net over F8, using
(159−16, 159, large)-Net in Base 8 — Upper bound on s
There is no (143, 159, large)-net in base 8, because
- 14 times m-reduction [i] would yield (143, 145, large)-net in base 8, but