Best Known (110−17, 110, s)-Nets in Base 7
(110−17, 110, 102957)-Net over F7 — Constructive and digital
Digital (93, 110, 102957)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 14)-net over F7, using
- 3 times m-reduction [i] based on digital (3, 14, 14)-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 (3, 11, 14)-net over F7, using
(110−17, 110, 823595)-Net over F7 — Digital
Digital (93, 110, 823595)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7110, 823595, F7, 17) (dual of [823595, 823485, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [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(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(711, 52, F7, 6) (dual of [52, 41, 7]-code), using
- construction XX applied to C1 = C({0,1,2,3,41}), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C({0,1,2,3,4,41}) [i] based on
- linear OA(79, 48, F7, 5) (dual of [48, 39, 6]-code), using the primitive cyclic code C(A) with length 48 = 72−1, defining set A = {0,1,2,3,41}, and minimum distance d ≥ |{−1,0,1,2,3}|+1 = 6 (BCH-bound) [i]
- linear OA(79, 48, F7, 5) (dual of [48, 39, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(711, 48, F7, 6) (dual of [48, 37, 7]-code), using the primitive cyclic code C(A) with length 48 = 72−1, defining set A = {0,1,2,3,4,41}, and minimum distance d ≥ |{−1,0,…,4}|+1 = 7 (BCH-bound) [i]
- linear OA(77, 48, F7, 4) (dual of [48, 41, 5]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [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(70, 2, F7, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C({0,1,2,3,41}), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C({0,1,2,3,4,41}) [i] based on
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
(110−17, 110, large)-Net in Base 7 — Upper bound on s
There is no (93, 110, large)-net in base 7, because
- 15 times m-reduction [i] would yield (93, 95, large)-net in base 7, but