Best Known (72, 72+21, s)-Nets in Base 8
(72, 72+21, 3278)-Net over F8 — Constructive and digital
Digital (72, 93, 3278)-net over F8, using
- net defined by OOA [i] based on linear OOA(893, 3278, F8, 21, 21) (dual of [(3278, 21), 68745, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(893, 32781, F8, 21) (dual of [32781, 32688, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(892, 32780, F8, 21) (dual of [32780, 32688, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(891, 32769, F8, 21) (dual of [32769, 32678, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(881, 32769, F8, 19) (dual of [32769, 32688, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(892, 32780, F8, 21) (dual of [32780, 32688, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(893, 32781, F8, 21) (dual of [32781, 32688, 22]-code), using
(72, 72+21, 26717)-Net over F8 — Digital
Digital (72, 93, 26717)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(893, 26717, F8, 21) (dual of [26717, 26624, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(893, 32781, F8, 21) (dual of [32781, 32688, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(892, 32780, F8, 21) (dual of [32780, 32688, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(891, 32769, F8, 21) (dual of [32769, 32678, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(881, 32769, F8, 19) (dual of [32769, 32688, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(892, 32780, F8, 21) (dual of [32780, 32688, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(893, 32781, F8, 21) (dual of [32781, 32688, 22]-code), using
(72, 72+21, large)-Net in Base 8 — Upper bound on s
There is no (72, 93, large)-net in base 8, because
- 19 times m-reduction [i] would yield (72, 74, large)-net in base 8, but