Best Known (69, 69+29, s)-Nets in Base 5
(69, 69+29, 296)-Net over F5 — Constructive and digital
Digital (69, 98, 296)-net over F5, using
- 2 times m-reduction [i] based on digital (69, 100, 296)-net over F5, using
- trace code for nets [i] based on digital (19, 50, 148)-net over F25, using
- net from sequence [i] based on digital (19, 147)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 19 and N(F) ≥ 148, using
- net from sequence [i] based on digital (19, 147)-sequence over F25, using
- trace code for nets [i] based on digital (19, 50, 148)-net over F25, using
(69, 69+29, 804)-Net over F5 — Digital
Digital (69, 98, 804)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(598, 804, F5, 29) (dual of [804, 706, 30]-code), using
- 163 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 12 times 0, 1, 26 times 0, 1, 35 times 0, 1, 39 times 0, 1, 42 times 0) [i] based on linear OA(592, 635, F5, 29) (dual of [635, 543, 30]-code), using
- construction XX applied to C1 = C([622,25]), C2 = C([0,26]), C3 = C1 + C2 = C([0,25]), and C∩ = C1 ∩ C2 = C([622,26]) [i] based on
- linear OA(589, 624, F5, 28) (dual of [624, 535, 29]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,25}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(583, 624, F5, 27) (dual of [624, 541, 28]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(591, 624, F5, 29) (dual of [624, 533, 30]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,26}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(581, 624, F5, 26) (dual of [624, 543, 27]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [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, 2, F5, 0) (dual of [2, 2, 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([622,25]), C2 = C([0,26]), C3 = C1 + C2 = C([0,25]), and C∩ = C1 ∩ C2 = C([622,26]) [i] based on
- 163 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 12 times 0, 1, 26 times 0, 1, 35 times 0, 1, 39 times 0, 1, 42 times 0) [i] based on linear OA(592, 635, F5, 29) (dual of [635, 543, 30]-code), using
(69, 69+29, 105249)-Net in Base 5 — Upper bound on s
There is no (69, 98, 105250)-net in base 5, because
- 1 times m-reduction [i] would yield (69, 97, 105250)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 63 112971 178167 161493 858941 121623 536704 384848 337182 675918 812081 692801 > 597 [i]