Best Known (16−4, 16, s)-Nets in Base 7
(16−4, 16, 8406)-Net over F7 — Constructive and digital
Digital (12, 16, 8406)-net over F7, using
- net defined by OOA [i] based on linear OOA(716, 8406, F7, 4, 4) (dual of [(8406, 4), 33608, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(716, 8406, F7, 3, 4) (dual of [(8406, 3), 25202, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(716, 16812, F7, 4) (dual of [16812, 16796, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(716, 16807, F7, 4) (dual of [16807, 16791, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(711, 16807, F7, 3) (dual of [16807, 16796, 4]-code or 16807-cap in PG(10,7)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(716, 16812, F7, 4) (dual of [16812, 16796, 5]-code), using
- appending kth column [i] based on linear OOA(716, 8406, F7, 3, 4) (dual of [(8406, 3), 25202, 5]-NRT-code), using
(16−4, 16, 16812)-Net over F7 — Digital
Digital (12, 16, 16812)-net over F7, using
- net defined by OOA [i] based on linear OOA(716, 16812, F7, 4, 4) (dual of [(16812, 4), 67232, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(716, 16812, F7, 3, 4) (dual of [(16812, 3), 50420, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(716, 16812, F7, 4) (dual of [16812, 16796, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(716, 16807, F7, 4) (dual of [16807, 16791, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(711, 16807, F7, 3) (dual of [16807, 16796, 4]-code or 16807-cap in PG(10,7)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(716, 16812, F7, 4) (dual of [16812, 16796, 5]-code), using
- appending kth column [i] based on linear OOA(716, 16812, F7, 3, 4) (dual of [(16812, 3), 50420, 5]-NRT-code), using
(16−4, 16, 1358775)-Net in Base 7 — Upper bound on s
There is no (12, 16, 1358776)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 33 232940 690449 > 716 [i]