Best Known (48−13, 48, s)-Nets in Base 5
(48−13, 48, 252)-Net over F5 — Constructive and digital
Digital (35, 48, 252)-net over F5, using
- 2 times m-reduction [i] based on digital (35, 50, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 25, 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, 25, 126)-net over F25, using
(48−13, 48, 843)-Net over F5 — Digital
Digital (35, 48, 843)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(548, 843, F5, 13) (dual of [843, 795, 14]-code), using
- 204 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 33 times 0, 1, 57 times 0, 1, 84 times 0) [i] based on linear OA(541, 632, F5, 13) (dual of [632, 591, 14]-code), using
- construction XX applied to C1 = C([623,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([623,11]) [i] based on
- linear OA(537, 624, F5, 12) (dual of [624, 587, 13]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(537, 624, F5, 12) (dual of [624, 587, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(541, 624, F5, 13) (dual of [624, 583, 14]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(533, 624, F5, 11) (dual of [624, 591, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [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([623,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([623,11]) [i] based on
- 204 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 33 times 0, 1, 57 times 0, 1, 84 times 0) [i] based on linear OA(541, 632, F5, 13) (dual of [632, 591, 14]-code), using
(48−13, 48, 223573)-Net in Base 5 — Upper bound on s
There is no (35, 48, 223574)-net in base 5, because
- 1 times m-reduction [i] would yield (35, 47, 223574)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 710 561794 342872 191721 360708 172097 > 547 [i]