Best Known (17−4, 17, s)-Nets in Base 7
(17−4, 17, 8409)-Net over F7 — Constructive and digital
Digital (13, 17, 8409)-net over F7, using
- net defined by OOA [i] based on linear OOA(717, 8409, F7, 4, 4) (dual of [(8409, 4), 33619, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(717, 8409, F7, 3, 4) (dual of [(8409, 3), 25210, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(717, 16818, F7, 4) (dual of [16818, 16801, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(1) [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(76, 16807, F7, 2) (dual of [16807, 16801, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 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(3) ⊂ Ce(1) [i] based on
- OA 2-folding and stacking [i] based on linear OA(717, 16818, F7, 4) (dual of [16818, 16801, 5]-code), using
- appending kth column [i] based on linear OOA(717, 8409, F7, 3, 4) (dual of [(8409, 3), 25210, 5]-NRT-code), using
(17−4, 17, 18627)-Net over F7 — Digital
Digital (13, 17, 18627)-net over F7, using
- net defined by OOA [i] based on linear OOA(717, 18627, F7, 4, 4) (dual of [(18627, 4), 74491, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(717, 18627, F7, 3, 4) (dual of [(18627, 3), 55864, 5]-NRT-code), using
(17−4, 17, 3594984)-Net in Base 7 — Upper bound on s
There is no (13, 17, 3594985)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 232 630616 553601 > 717 [i]