Best Known (85, 110, s)-Nets in Base 7
(85, 110, 1402)-Net over F7 — Constructive and digital
Digital (85, 110, 1402)-net over F7, using
- 71 times duplication [i] based on digital (84, 109, 1402)-net over F7, using
- net defined by OOA [i] based on linear OOA(7109, 1402, F7, 25, 25) (dual of [(1402, 25), 34941, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(7109, 16825, F7, 25) (dual of [16825, 16716, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(7106, 16807, F7, 25) (dual of [16807, 16701, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(791, 16807, F7, 22) (dual of [16807, 16716, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(73, 18, F7, 2) (dual of [18, 15, 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(24) ⊂ Ce(21) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(7109, 16825, F7, 25) (dual of [16825, 16716, 26]-code), using
- net defined by OOA [i] based on linear OOA(7109, 1402, F7, 25, 25) (dual of [(1402, 25), 34941, 26]-NRT-code), using
(85, 110, 15883)-Net over F7 — Digital
Digital (85, 110, 15883)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7110, 15883, F7, 25) (dual of [15883, 15773, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(7110, 16826, F7, 25) (dual of [16826, 16716, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(7109, 16825, F7, 25) (dual of [16825, 16716, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(7106, 16807, F7, 25) (dual of [16807, 16701, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(791, 16807, F7, 22) (dual of [16807, 16716, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(73, 18, F7, 2) (dual of [18, 15, 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(24) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(7109, 16825, F7, 25) (dual of [16825, 16716, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(7110, 16826, F7, 25) (dual of [16826, 16716, 26]-code), using
(85, 110, large)-Net in Base 7 — Upper bound on s
There is no (85, 110, large)-net in base 7, because
- 23 times m-reduction [i] would yield (85, 87, large)-net in base 7, but