Best Known (20, 25, s)-Nets in Base 7
(20, 25, 58827)-Net over F7 — Constructive and digital
Digital (20, 25, 58827)-net over F7, using
- net defined by OOA [i] based on linear OOA(725, 58827, F7, 5, 5) (dual of [(58827, 5), 294110, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(725, 117655, F7, 5) (dual of [117655, 117630, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(725, 117649, F7, 5) (dual of [117649, 117624, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(719, 117649, F7, 4) (dual of [117649, 117630, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(70, 6, F7, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(725, 117655, F7, 5) (dual of [117655, 117630, 6]-code), using
(20, 25, 117655)-Net over F7 — Digital
Digital (20, 25, 117655)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(725, 117655, F7, 5) (dual of [117655, 117630, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(725, 117649, F7, 5) (dual of [117649, 117624, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(719, 117649, F7, 4) (dual of [117649, 117630, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(70, 6, F7, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
(20, 25, large)-Net in Base 7 — Upper bound on s
There is no (20, 25, large)-net in base 7, because
- 3 times m-reduction [i] would yield (20, 22, large)-net in base 7, but