Best Known (36−10, 36, s)-Nets in Base 7
(36−10, 36, 482)-Net over F7 — Constructive and digital
Digital (26, 36, 482)-net over F7, using
- 72 times duplication [i] based on digital (24, 34, 482)-net over F7, using
- net defined by OOA [i] based on linear OOA(734, 482, F7, 10, 10) (dual of [(482, 10), 4786, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(734, 2410, F7, 10) (dual of [2410, 2376, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(733, 2401, F7, 10) (dual of [2401, 2368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(71, 9, F7, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- OA 5-folding and stacking [i] based on linear OA(734, 2410, F7, 10) (dual of [2410, 2376, 11]-code), using
- net defined by OOA [i] based on linear OOA(734, 482, F7, 10, 10) (dual of [(482, 10), 4786, 11]-NRT-code), using
(36−10, 36, 2446)-Net over F7 — Digital
Digital (26, 36, 2446)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(736, 2446, F7, 10) (dual of [2446, 2410, 11]-code), using
- 38 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 31 times 0) [i] based on linear OA(733, 2405, F7, 10) (dual of [2405, 2372, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(733, 2401, F7, 10) (dual of [2401, 2368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(729, 2401, F7, 9) (dual of [2401, 2372, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 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
- 38 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 31 times 0) [i] based on linear OA(733, 2405, F7, 10) (dual of [2405, 2372, 11]-code), using
(36−10, 36, 527701)-Net in Base 7 — Upper bound on s
There is no (26, 36, 527702)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 2 651736 494836 625698 576051 886077 > 736 [i]