Best Known (69, 69+19, s)-Nets in Base 5
(69, 69+19, 411)-Net over F5 — Constructive and digital
Digital (69, 88, 411)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (21, 30, 159)-net over F5, using
- net defined by OOA [i] based on linear OOA(530, 159, F5, 9, 9) (dual of [(159, 9), 1401, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(530, 637, F5, 9) (dual of [637, 607, 10]-code), using
- construction XX applied to C1 = C([622,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([622,6]) [i] based on
- linear OA(525, 624, F5, 8) (dual of [624, 599, 9]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,5}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(521, 624, F5, 7) (dual of [624, 603, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(529, 624, F5, 9) (dual of [624, 595, 10]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(517, 624, F5, 6) (dual of [624, 607, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 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
- 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 XX applied to C1 = C([622,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([622,6]) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(530, 637, F5, 9) (dual of [637, 607, 10]-code), using
- net defined by OOA [i] based on linear OOA(530, 159, F5, 9, 9) (dual of [(159, 9), 1401, 10]-NRT-code), using
- 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
- digital (21, 30, 159)-net over F5, using
(69, 69+19, 4944)-Net over F5 — Digital
Digital (69, 88, 4944)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(588, 4944, F5, 19) (dual of [4944, 4856, 20]-code), using
- 4855 step Varšamov–Edel lengthening with (ri) = (4, 3, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 15 times 0, 1, 17 times 0, 1, 18 times 0, 1, 20 times 0, 1, 22 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 33 times 0, 1, 35 times 0, 1, 40 times 0, 1, 43 times 0, 1, 47 times 0, 1, 52 times 0, 1, 57 times 0, 1, 63 times 0, 1, 68 times 0, 1, 75 times 0, 1, 83 times 0, 1, 90 times 0, 1, 99 times 0, 1, 108 times 0, 1, 119 times 0, 1, 130 times 0, 1, 142 times 0, 1, 156 times 0, 1, 171 times 0, 1, 187 times 0, 1, 204 times 0, 1, 224 times 0, 1, 245 times 0, 1, 268 times 0, 1, 293 times 0, 1, 321 times 0, 1, 351 times 0, 1, 384 times 0, 1, 420 times 0) [i] based on linear OA(519, 20, F5, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,5)), using
- dual of repetition code with length 20 [i]
- 4855 step Varšamov–Edel lengthening with (ri) = (4, 3, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 15 times 0, 1, 17 times 0, 1, 18 times 0, 1, 20 times 0, 1, 22 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 33 times 0, 1, 35 times 0, 1, 40 times 0, 1, 43 times 0, 1, 47 times 0, 1, 52 times 0, 1, 57 times 0, 1, 63 times 0, 1, 68 times 0, 1, 75 times 0, 1, 83 times 0, 1, 90 times 0, 1, 99 times 0, 1, 108 times 0, 1, 119 times 0, 1, 130 times 0, 1, 142 times 0, 1, 156 times 0, 1, 171 times 0, 1, 187 times 0, 1, 204 times 0, 1, 224 times 0, 1, 245 times 0, 1, 268 times 0, 1, 293 times 0, 1, 321 times 0, 1, 351 times 0, 1, 384 times 0, 1, 420 times 0) [i] based on linear OA(519, 20, F5, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,5)), using
(69, 69+19, 5921081)-Net in Base 5 — Upper bound on s
There is no (69, 88, 5921082)-net in base 5, because
- 1 times m-reduction [i] would yield (69, 87, 5921082)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 6 462353 266302 229369 034282 635155 420539 817454 760583 073836 418025 > 587 [i]