Best Known (21, 21+6, s)-Nets in Base 8
(21, 21+6, 10926)-Net over F8 — Constructive and digital
Digital (21, 27, 10926)-net over F8, using
- net defined by OOA [i] based on linear OOA(827, 10926, F8, 6, 6) (dual of [(10926, 6), 65529, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(827, 32778, F8, 6) (dual of [32778, 32751, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(827, 32779, F8, 6) (dual of [32779, 32752, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- 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(816, 32768, F8, 4) (dual of [32768, 32752, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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 Ce(5) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(827, 32779, F8, 6) (dual of [32779, 32752, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(827, 32778, F8, 6) (dual of [32778, 32751, 7]-code), using
(21, 21+6, 32780)-Net over F8 — Digital
Digital (21, 27, 32780)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(827, 32780, F8, 6) (dual of [32780, 32753, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- 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(816, 32768, F8, 4) (dual of [32768, 32752, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(5) ⊂ Ce(3) [i] based on
(21, 21+6, large)-Net in Base 8 — Upper bound on s
There is no (21, 27, large)-net in base 8, because
- 4 times m-reduction [i] would yield (21, 23, large)-net in base 8, but