Best Known (72−18, 72, s)-Nets in Base 7
(72−18, 72, 534)-Net over F7 — Constructive and digital
Digital (54, 72, 534)-net over F7, using
- 72 times duplication [i] based on digital (52, 70, 534)-net over F7, using
- net defined by OOA [i] based on linear OOA(770, 534, F7, 18, 18) (dual of [(534, 18), 9542, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(770, 4806, F7, 18) (dual of [4806, 4736, 19]-code), using
- trace code [i] based on linear OA(4935, 2403, F49, 18) (dual of [2403, 2368, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(4935, 2401, F49, 18) (dual of [2401, 2366, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4933, 2401, F49, 17) (dual of [2401, 2368, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(490, 2, F49, 0) (dual of [2, 2, 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(17) ⊂ Ce(16) [i] based on
- trace code [i] based on linear OA(4935, 2403, F49, 18) (dual of [2403, 2368, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(770, 4806, F7, 18) (dual of [4806, 4736, 19]-code), using
- net defined by OOA [i] based on linear OOA(770, 534, F7, 18, 18) (dual of [(534, 18), 9542, 19]-NRT-code), using
(72−18, 72, 4899)-Net over F7 — Digital
Digital (54, 72, 4899)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(772, 4899, F7, 18) (dual of [4899, 4827, 19]-code), using
- 91 step Varšamov–Edel lengthening with (ri) = (1, 12 times 0, 1, 77 times 0) [i] based on linear OA(770, 4806, F7, 18) (dual of [4806, 4736, 19]-code), using
- trace code [i] based on linear OA(4935, 2403, F49, 18) (dual of [2403, 2368, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(4935, 2401, F49, 18) (dual of [2401, 2366, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4933, 2401, F49, 17) (dual of [2401, 2368, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(490, 2, F49, 0) (dual of [2, 2, 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(17) ⊂ Ce(16) [i] based on
- trace code [i] based on linear OA(4935, 2403, F49, 18) (dual of [2403, 2368, 19]-code), using
- 91 step Varšamov–Edel lengthening with (ri) = (1, 12 times 0, 1, 77 times 0) [i] based on linear OA(770, 4806, F7, 18) (dual of [4806, 4736, 19]-code), using
(72−18, 72, 3984592)-Net in Base 7 — Upper bound on s
There is no (54, 72, 3984593)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 7 031683 663500 861100 785784 036802 755118 043421 633977 119055 060775 > 772 [i]