Best Known (32, 43, s)-Nets in Base 7
(32, 43, 961)-Net over F7 — Constructive and digital
Digital (32, 43, 961)-net over F7, using
- 71 times duplication [i] based on digital (31, 42, 961)-net over F7, using
- net defined by OOA [i] based on linear OOA(742, 961, F7, 11, 11) (dual of [(961, 11), 10529, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(742, 4806, F7, 11) (dual of [4806, 4764, 12]-code), using
- trace code [i] based on linear OA(4921, 2403, F49, 11) (dual of [2403, 2382, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(4921, 2401, F49, 11) (dual of [2401, 2380, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(4919, 2401, F49, 10) (dual of [2401, 2382, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(4921, 2403, F49, 11) (dual of [2403, 2382, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(742, 4806, F7, 11) (dual of [4806, 4764, 12]-code), using
- net defined by OOA [i] based on linear OOA(742, 961, F7, 11, 11) (dual of [(961, 11), 10529, 12]-NRT-code), using
(32, 43, 4815)-Net over F7 — Digital
Digital (32, 43, 4815)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(743, 4815, F7, 11) (dual of [4815, 4772, 12]-code), using
- 8 step Varšamov–Edel lengthening with (ri) = (1, 7 times 0) [i] based on linear OA(742, 4806, F7, 11) (dual of [4806, 4764, 12]-code), using
- trace code [i] based on linear OA(4921, 2403, F49, 11) (dual of [2403, 2382, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(4921, 2401, F49, 11) (dual of [2401, 2380, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(4919, 2401, F49, 10) (dual of [2401, 2382, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(4921, 2403, F49, 11) (dual of [2403, 2382, 12]-code), using
- 8 step Varšamov–Edel lengthening with (ri) = (1, 7 times 0) [i] based on linear OA(742, 4806, F7, 11) (dual of [4806, 4764, 12]-code), using
(32, 43, 5451403)-Net in Base 7 — Upper bound on s
There is no (32, 43, 5451404)-net in base 7, because
- 1 times m-reduction [i] would yield (32, 42, 5451404)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 311973 695193 919816 416721 907107 418857 > 742 [i]