Best Known (108−17, 108, s)-Nets in Base 7
(108−17, 108, 102956)-Net over F7 — Constructive and digital
Digital (91, 108, 102956)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 13)-net over F7, using
- 4 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- 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
- net defined by OOA [i] based on linear OOA(799, 102943, F7, 17, 17) (dual of [(102943, 17), 1749932, 18]-NRT-code), using
- digital (1, 9, 13)-net over F7, using
(108−17, 108, 823589)-Net over F7 — Digital
Digital (91, 108, 823589)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7108, 823589, F7, 17) (dual of [823589, 823481, 18]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(7106, 823585, F7, 17) (dual of [823585, 823479, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [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(764, 823543, F7, 11) (dual of [823543, 823479, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(77, 42, F7, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(7106, 823587, F7, 16) (dual of [823587, 823481, 17]-code), using Gilbert–Varšamov bound and bm = 7106 > Vbs−1(k−1) = 19559 125194 446379 127167 724686 002434 909544 756742 544585 628241 271625 329874 754495 971051 053873 [i]
- linear OA(70, 2, F7, 0) (dual of [2, 2, 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
- linear OA(7106, 823585, F7, 17) (dual of [823585, 823479, 18]-code), using
- construction X with Varšamov bound [i] based on
(108−17, 108, large)-Net in Base 7 — Upper bound on s
There is no (91, 108, large)-net in base 7, because
- 15 times m-reduction [i] would yield (91, 93, large)-net in base 7, but