Best Known (48, 48+18, s)-Nets in Base 5
(48, 48+18, 262)-Net over F5 — Constructive and digital
Digital (48, 66, 262)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 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
- a shift-net [i]
- net from sequence [i] based on digital (1, 9)-sequence over F5, 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 (1, 10, 10)-net over F5, using
(48, 48+18, 937)-Net over F5 — Digital
Digital (48, 66, 937)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(566, 937, F5, 18) (dual of [937, 871, 19]-code), using
- 296 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 11 times 0, 1, 24 times 0, 1, 40 times 0, 1, 57 times 0, 1, 70 times 0, 1, 80 times 0) [i] based on linear OA(557, 632, F5, 18) (dual of [632, 575, 19]-code), using
- construction XX applied to C1 = C([623,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([623,16]) [i] based on
- linear OA(553, 624, F5, 17) (dual of [624, 571, 18]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,15}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(553, 624, F5, 17) (dual of [624, 571, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(557, 624, F5, 18) (dual of [624, 567, 19]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(549, 624, F5, 16) (dual of [624, 575, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [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
- linear OA(50, 4, F5, 0) (dual of [4, 4, 1]-code) (see above)
- construction XX applied to C1 = C([623,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([623,16]) [i] based on
- 296 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 11 times 0, 1, 24 times 0, 1, 40 times 0, 1, 57 times 0, 1, 70 times 0, 1, 80 times 0) [i] based on linear OA(557, 632, F5, 18) (dual of [632, 575, 19]-code), using
(48, 48+18, 138500)-Net in Base 5 — Upper bound on s
There is no (48, 66, 138501)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 13552 812992 648921 917240 087004 931781 478706 379125 > 566 [i]