Best Known (104−16, 104, s)-Nets in Base 7
(104−16, 104, 102959)-Net over F7 — Constructive and digital
Digital (88, 104, 102959)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 16)-net over F7, using
- 4 times m-reduction [i] based on digital (4, 16, 16)-net over F7, using
- digital (76, 92, 102943)-net over F7, using
- net defined by OOA [i] based on linear OOA(792, 102943, F7, 16, 16) (dual of [(102943, 16), 1646996, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(792, 823544, F7, 16) (dual of [823544, 823452, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(792, 823550, F7, 16) (dual of [823550, 823458, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- 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(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(15) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(792, 823550, F7, 16) (dual of [823550, 823458, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(792, 823544, F7, 16) (dual of [823544, 823452, 17]-code), using
- net defined by OOA [i] based on linear OOA(792, 102943, F7, 16, 16) (dual of [(102943, 16), 1646996, 17]-NRT-code), using
- digital (4, 12, 16)-net over F7, using
(104−16, 104, 823597)-Net over F7 — Digital
Digital (88, 104, 823597)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7104, 823597, F7, 16) (dual of [823597, 823493, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(8) [i] based on
- 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(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(712, 54, F7, 6) (dual of [54, 42, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(712, 100, F7, 6) (dual of [100, 88, 7]-code), using
- trace code [i] based on linear OA(496, 50, F49, 6) (dual of [50, 44, 7]-code or 50-arc in PG(5,49)), using
- extended Reed–Solomon code RSe(44,49) [i]
- algebraic-geometric code AG(F, Q+20P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using the rational function field F49(x) [i]
- algebraic-geometric code AG(F, Q+13P) with degQ = 4 and degPÂ =Â 3 [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50 (see above)
- algebraic-geometric code AG(F, Q+8P) with degQ = 3 and degPÂ =Â 5 [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50 (see above)
- trace code [i] based on linear OA(496, 50, F49, 6) (dual of [50, 44, 7]-code or 50-arc in PG(5,49)), using
- discarding factors / shortening the dual code based on linear OA(712, 100, F7, 6) (dual of [100, 88, 7]-code), using
- construction X applied to Ce(15) ⊂ Ce(8) [i] based on
(104−16, 104, large)-Net in Base 7 — Upper bound on s
There is no (88, 104, large)-net in base 7, because
- 14 times m-reduction [i] would yield (88, 90, large)-net in base 7, but