Best Known (54, 54+22, s)-Nets in Base 5
(54, 54+22, 262)-Net over F5 — Constructive and digital
Digital (54, 76, 262)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (42, 64, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 32, 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, 32, 126)-net over F25, using
- digital (1, 12, 10)-net over F5, using
(54, 54+22, 749)-Net over F5 — Digital
Digital (54, 76, 749)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(576, 749, F5, 22) (dual of [749, 673, 23]-code), using
- 113 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0, 1, 18 times 0, 1, 31 times 0, 1, 44 times 0) [i] based on linear OA(569, 629, F5, 22) (dual of [629, 560, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(569, 625, F5, 22) (dual of [625, 556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(565, 625, F5, 21) (dual of [625, 560, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(50, 4, F5, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 113 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0, 1, 18 times 0, 1, 31 times 0, 1, 44 times 0) [i] based on linear OA(569, 629, F5, 22) (dual of [629, 560, 23]-code), using
(54, 54+22, 82824)-Net in Base 5 — Upper bound on s
There is no (54, 76, 82825)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 132350 946147 355957 487250 738907 180029 490560 885650 146941 > 576 [i]