Best Known (41−10, 41, s)-Nets in Base 7
(41−10, 41, 3362)-Net over F7 — Constructive and digital
Digital (31, 41, 3362)-net over F7, using
- net defined by OOA [i] based on linear OOA(741, 3362, F7, 10, 10) (dual of [(3362, 10), 33579, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(741, 16810, F7, 10) (dual of [16810, 16769, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(741, 16812, F7, 10) (dual of [16812, 16771, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(736, 16807, F7, 9) (dual of [16807, 16771, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(70, 5, F7, 0) (dual of [5, 5, 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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(741, 16812, F7, 10) (dual of [16812, 16771, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(741, 16810, F7, 10) (dual of [16810, 16769, 11]-code), using
(41−10, 41, 10540)-Net over F7 — Digital
Digital (31, 41, 10540)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(741, 10540, F7, 10) (dual of [10540, 10499, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using
(41−10, 41, 3693929)-Net in Base 7 — Upper bound on s
There is no (31, 41, 3693930)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 44567 674945 486604 105288 509312 880437 > 741 [i]