Best Known (62, 62+18, s)-Nets in Base 5
(62, 62+18, 356)-Net over F5 — Constructive and digital
Digital (62, 80, 356)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (15, 24, 104)-net over F5, using
- trace code for nets [i] based on digital (3, 12, 52)-net over F25, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- trace code for nets [i] based on digital (3, 12, 52)-net over F25, using
- digital (38, 56, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 28, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 28, 126)-net over F25, using
- digital (15, 24, 104)-net over F5, using
(62, 62+18, 3589)-Net over F5 — Digital
Digital (62, 80, 3589)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(580, 3589, F5, 18) (dual of [3589, 3509, 19]-code), using
- 445 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 13 times 0, 1, 30 times 0, 1, 63 times 0, 1, 122 times 0, 1, 204 times 0) [i] based on linear OA(572, 3136, F5, 18) (dual of [3136, 3064, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 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
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- 445 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 13 times 0, 1, 30 times 0, 1, 63 times 0, 1, 122 times 0, 1, 204 times 0) [i] based on linear OA(572, 3136, F5, 18) (dual of [3136, 3064, 19]-code), using
(62, 62+18, 1693388)-Net in Base 5 — Upper bound on s
There is no (62, 80, 1693389)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 82 718361 673956 161027 076638 638822 845966 093043 330090 239125 > 580 [i]