Best Known (69−15, 69, s)-Nets in Base 5
(69−15, 69, 458)-Net over F5 — Constructive and digital
Digital (54, 69, 458)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- digital (45, 60, 446)-net over F5, using
- net defined by OOA [i] based on linear OOA(560, 446, F5, 15, 15) (dual of [(446, 15), 6630, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(560, 3123, F5, 15) (dual of [3123, 3063, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(560, 3124, F5, 15) (dual of [3124, 3064, 16]-code), using
- 1 times truncation [i] based on linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 1 times truncation [i] based on linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(560, 3124, F5, 15) (dual of [3124, 3064, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(560, 3123, F5, 15) (dual of [3123, 3063, 16]-code), using
- net defined by OOA [i] based on linear OOA(560, 446, F5, 15, 15) (dual of [(446, 15), 6630, 16]-NRT-code), using
- digital (2, 9, 12)-net over F5, using
(69−15, 69, 4243)-Net over F5 — Digital
Digital (54, 69, 4243)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(569, 4243, F5, 15) (dual of [4243, 4174, 16]-code), using
- 1110 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 23 times 0, 1, 52 times 0, 1, 107 times 0, 1, 195 times 0, 1, 305 times 0, 1, 409 times 0) [i] based on linear OA(560, 3124, F5, 15) (dual of [3124, 3064, 16]-code), using
- 1 times truncation [i] based on linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 1 times truncation [i] based on linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using
- 1110 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 23 times 0, 1, 52 times 0, 1, 107 times 0, 1, 195 times 0, 1, 305 times 0, 1, 409 times 0) [i] based on linear OA(560, 3124, F5, 15) (dual of [3124, 3064, 16]-code), using
(69−15, 69, 5210178)-Net in Base 5 — Upper bound on s
There is no (54, 69, 5210179)-net in base 5, because
- 1 times m-reduction [i] would yield (54, 68, 5210179)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 338813 481802 270477 532360 915948 242008 183014 827445 > 568 [i]