Best Known (109, 143, s)-Nets in Base 3
(109, 143, 400)-Net over F3 — Constructive and digital
Digital (109, 143, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (109, 144, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 36, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 36, 100)-net over F81, using
(109, 143, 803)-Net over F3 — Digital
Digital (109, 143, 803)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3143, 803, F3, 34) (dual of [803, 660, 35]-code), using
- 50 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 14 times 0) [i] based on linear OA(3132, 742, F3, 34) (dual of [742, 610, 35]-code), using
- construction XX applied to C1 = C([336,367]), C2 = C([334,365]), C3 = C1 + C2 = C([336,365]), and C∩ = C1 ∩ C2 = C([334,367]) [i] based on
- linear OA(3124, 728, F3, 32) (dual of [728, 604, 33]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {336,337,…,367}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3124, 728, F3, 32) (dual of [728, 604, 33]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {334,335,…,365}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3130, 728, F3, 34) (dual of [728, 598, 35]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {334,335,…,367}, and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3118, 728, F3, 30) (dual of [728, 610, 31]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {336,337,…,365}, and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(31, 7, F3, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(31, 7, F3, 1) (dual of [7, 6, 2]-code) (see above)
- construction XX applied to C1 = C([336,367]), C2 = C([334,365]), C3 = C1 + C2 = C([336,365]), and C∩ = C1 ∩ C2 = C([334,367]) [i] based on
- 50 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 14 times 0) [i] based on linear OA(3132, 742, F3, 34) (dual of [742, 610, 35]-code), using
(109, 143, 36995)-Net in Base 3 — Upper bound on s
There is no (109, 143, 36996)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 169 185813 383951 780760 924306 798407 421521 350477 934107 401653 699310 858249 > 3143 [i]