Best Known (109−25, 109, s)-Nets in Base 7
(109−25, 109, 1402)-Net over F7 — Constructive and digital
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
(109−25, 109, 14594)-Net over F7 — Digital
Digital (84, 109, 14594)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7109, 14594, F7, 25) (dual of [14594, 14485, 26]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(7109, 16825, F7, 25) (dual of [16825, 16716, 26]-code), using
(109−25, 109, large)-Net in Base 7 — Upper bound on s
There is no (84, 109, large)-net in base 7, because
- 23 times m-reduction [i] would yield (84, 86, large)-net in base 7, but