Best Known (91, 110, s)-Nets in Base 7
(91, 110, 26144)-Net over F7 — Constructive and digital
Digital (91, 110, 26144)-net over F7, using
- net defined by OOA [i] based on linear OOA(7110, 26144, F7, 19, 19) (dual of [(26144, 19), 496626, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(7110, 235297, F7, 19) (dual of [235297, 235187, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(7110, 235300, F7, 19) (dual of [235300, 235190, 20]-code), using
- trace code [i] based on linear OA(4955, 117650, F49, 19) (dual of [117650, 117595, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- trace code [i] based on linear OA(4955, 117650, F49, 19) (dual of [117650, 117595, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(7110, 235300, F7, 19) (dual of [235300, 235190, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(7110, 235297, F7, 19) (dual of [235297, 235187, 20]-code), using
(91, 110, 235304)-Net over F7 — Digital
Digital (91, 110, 235304)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7110, 235304, F7, 19) (dual of [235304, 235194, 20]-code), using
- trace code [i] based on linear OA(4955, 117652, F49, 19) (dual of [117652, 117597, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(4955, 117649, F49, 19) (dual of [117649, 117594, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(4952, 117649, F49, 18) (dual of [117649, 117597, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(490, 3, F49, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- trace code [i] based on linear OA(4955, 117652, F49, 19) (dual of [117652, 117597, 20]-code), using
(91, 110, large)-Net in Base 7 — Upper bound on s
There is no (91, 110, large)-net in base 7, because
- 17 times m-reduction [i] would yield (91, 93, large)-net in base 7, but