Best Known (48−11, 48, s)-Nets in Base 8
(48−11, 48, 6556)-Net over F8 — Constructive and digital
Digital (37, 48, 6556)-net over F8, using
- net defined by OOA [i] based on linear OOA(848, 6556, F8, 11, 11) (dual of [(6556, 11), 72068, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(848, 32781, F8, 11) (dual of [32781, 32733, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(847, 32780, F8, 11) (dual of [32780, 32733, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(846, 32768, F8, 11) (dual of [32768, 32722, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(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(10) ⊂ Ce(8) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(847, 32780, F8, 11) (dual of [32780, 32733, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(848, 32781, F8, 11) (dual of [32781, 32733, 12]-code), using
(48−11, 48, 30812)-Net over F8 — Digital
Digital (37, 48, 30812)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(848, 30812, F8, 11) (dual of [30812, 30764, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(848, 32781, F8, 11) (dual of [32781, 32733, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(847, 32780, F8, 11) (dual of [32780, 32733, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(846, 32768, F8, 11) (dual of [32768, 32722, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(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(10) ⊂ Ce(8) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(847, 32780, F8, 11) (dual of [32780, 32733, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(848, 32781, F8, 11) (dual of [32781, 32733, 12]-code), using
(48−11, 48, large)-Net in Base 8 — Upper bound on s
There is no (37, 48, large)-net in base 8, because
- 9 times m-reduction [i] would yield (37, 39, large)-net in base 8, but