Best Known (99−17, 99, s)-Nets in Base 7
(99−17, 99, 102943)-Net over F7 — Constructive and digital
Digital (82, 99, 102943)-net over F7, using
- net defined by OOA [i] based on linear OOA(799, 102943, F7, 17, 17) (dual of [(102943, 17), 1749932, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(799, 823545, F7, 17) (dual of [823545, 823446, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(799, 823550, F7, 17) (dual of [823550, 823451, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(799, 823543, F7, 17) (dual of [823543, 823444, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(792, 823543, F7, 16) (dual of [823543, 823451, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 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(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(799, 823550, F7, 17) (dual of [823550, 823451, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(799, 823545, F7, 17) (dual of [823545, 823446, 18]-code), using
(99−17, 99, 411775)-Net over F7 — Digital
Digital (82, 99, 411775)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(799, 411775, F7, 2, 17) (dual of [(411775, 2), 823451, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(799, 823550, F7, 17) (dual of [823550, 823451, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(799, 823543, F7, 17) (dual of [823543, 823444, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(792, 823543, F7, 16) (dual of [823543, 823451, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 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(16) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(799, 823550, F7, 17) (dual of [823550, 823451, 18]-code), using
(99−17, 99, large)-Net in Base 7 — Upper bound on s
There is no (82, 99, large)-net in base 7, because
- 15 times m-reduction [i] would yield (82, 84, large)-net in base 7, but