Best Known (28, 36, s)-Nets in Base 7
(28, 36, 4214)-Net over F7 — Constructive and digital
Digital (28, 36, 4214)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 13)-net over F7, using
- 8 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (23, 31, 4201)-net over F7, using
- net defined by OOA [i] based on linear OOA(731, 4201, F7, 8, 8) (dual of [(4201, 8), 33577, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(731, 16804, F7, 8) (dual of [16804, 16773, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(731, 16804, F7, 8) (dual of [16804, 16773, 9]-code), using
- net defined by OOA [i] based on linear OOA(731, 4201, F7, 8, 8) (dual of [(4201, 8), 33577, 9]-NRT-code), using
- digital (1, 5, 13)-net over F7, using
(28, 36, 16828)-Net over F7 — Digital
Digital (28, 36, 16828)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(736, 16828, F7, 8) (dual of [16828, 16792, 9]-code), using
- construction XX applied to Ce(7) ⊂ Ce(3) ⊂ Ce(2) [i] based on
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(716, 16807, F7, 4) (dual of [16807, 16791, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(711, 16807, F7, 3) (dual of [16807, 16796, 4]-code or 16807-cap in PG(10,7)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(74, 20, F7, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,7)), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(7) ⊂ Ce(3) ⊂ Ce(2) [i] based on
(28, 36, large)-Net in Base 7 — Upper bound on s
There is no (28, 36, large)-net in base 7, because
- 6 times m-reduction [i] would yield (28, 30, large)-net in base 7, but