Best Known (108−20, 108, s)-Nets in Base 7
(108−20, 108, 11767)-Net over F7 — Constructive and digital
Digital (88, 108, 11767)-net over F7, using
- 72 times duplication [i] based on digital (86, 106, 11767)-net over F7, using
- net defined by OOA [i] based on linear OOA(7106, 11767, F7, 20, 20) (dual of [(11767, 20), 235234, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(7106, 117670, F7, 20) (dual of [117670, 117564, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(7103, 117649, F7, 20) (dual of [117649, 117546, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(785, 117649, F7, 17) (dual of [117649, 117564, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(73, 21, F7, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- OA 10-folding and stacking [i] based on linear OA(7106, 117670, F7, 20) (dual of [117670, 117564, 21]-code), using
- net defined by OOA [i] based on linear OOA(7106, 11767, F7, 20, 20) (dual of [(11767, 20), 235234, 21]-NRT-code), using
(108−20, 108, 117679)-Net over F7 — Digital
Digital (88, 108, 117679)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7108, 117679, F7, 20) (dual of [117679, 117571, 21]-code), using
- construction XX applied to Ce(19) ⊂ Ce(15) ⊂ Ce(14) [i] based on
- linear OA(7103, 117649, F7, 20) (dual of [117649, 117546, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(779, 117649, F7, 16) (dual of [117649, 117570, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(773, 117649, F7, 15) (dual of [117649, 117576, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(74, 29, F7, 3) (dual of [29, 25, 4]-code or 29-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(19) ⊂ Ce(15) ⊂ Ce(14) [i] based on
(108−20, 108, large)-Net in Base 7 — Upper bound on s
There is no (88, 108, large)-net in base 7, because
- 18 times m-reduction [i] would yield (88, 90, large)-net in base 7, but