Best Known (53−14, 53, s)-Nets in Base 5
(53−14, 53, 252)-Net over F5 — Constructive and digital
Digital (39, 53, 252)-net over F5, using
- 5 times m-reduction [i] based on digital (39, 58, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 29, 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, 29, 126)-net over F25, using
(53−14, 53, 1010)-Net over F5 — Digital
Digital (39, 53, 1010)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(553, 1010, F5, 14) (dual of [1010, 957, 15]-code), using
- 370 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 10 times 0, 1, 26 times 0, 1, 47 times 0, 1, 71 times 0, 1, 94 times 0, 1, 112 times 0) [i] based on linear OA(545, 632, F5, 14) (dual of [632, 587, 15]-code), using
- construction XX applied to C1 = C([623,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([623,12]) [i] based on
- 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(541, 624, F5, 13) (dual of [624, 583, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(545, 624, F5, 14) (dual of [624, 579, 15]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [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(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([623,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([623,12]) [i] based on
- 370 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 10 times 0, 1, 26 times 0, 1, 47 times 0, 1, 71 times 0, 1, 94 times 0, 1, 112 times 0) [i] based on linear OA(545, 632, F5, 14) (dual of [632, 587, 15]-code), using
(53−14, 53, 165595)-Net in Base 5 — Upper bound on s
There is no (39, 53, 165596)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 11 102620 551358 525812 429643 497680 843505 > 553 [i]