Best Known (67, 83, s)-Nets in Base 5
(67, 83, 1965)-Net over F5 — Constructive and digital
Digital (67, 83, 1965)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- digital (57, 73, 1953)-net over F5, using
- net defined by OOA [i] based on linear OOA(573, 1953, F5, 16, 16) (dual of [(1953, 16), 31175, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(573, 15624, F5, 16) (dual of [15624, 15551, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(573, 15624, F5, 16) (dual of [15624, 15551, 17]-code), using
- net defined by OOA [i] based on linear OOA(573, 1953, F5, 16, 16) (dual of [(1953, 16), 31175, 17]-NRT-code), using
- digital (2, 10, 12)-net over F5, using
(67, 83, 15662)-Net over F5 — Digital
Digital (67, 83, 15662)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(583, 15662, F5, 16) (dual of [15662, 15579, 17]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(579, 15655, F5, 16) (dual of [15655, 15576, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(579, 15658, F5, 14) (dual of [15658, 15579, 15]-code), using Gilbert–Varšamov bound and bm = 579 > Vbs−1(k−1) = 36444 905124 286882 983875 522980 917687 265790 301589 341925 [i]
- linear OA(51, 4, F5, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(579, 15655, F5, 16) (dual of [15655, 15576, 17]-code), using
- construction X with Varšamov bound [i] based on
(67, 83, large)-Net in Base 5 — Upper bound on s
There is no (67, 83, large)-net in base 5, because
- 14 times m-reduction [i] would yield (67, 69, large)-net in base 5, but