Best Known (37, 54, s)-Nets in Base 4
(37, 54, 240)-Net over F4 — Constructive and digital
Digital (37, 54, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 18, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
(37, 54, 276)-Net over F4 — Digital
Digital (37, 54, 276)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(454, 276, F4, 17) (dual of [276, 222, 18]-code), using
- construction XX applied to C1 = C([251,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([251,12]) [i] based on
- linear OA(445, 255, F4, 15) (dual of [255, 210, 16]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−4,−3,…,10}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(437, 255, F4, 13) (dual of [255, 218, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(449, 255, F4, 17) (dual of [255, 206, 18]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−4,−3,…,12}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(433, 255, F4, 11) (dual of [255, 222, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(44, 16, F4, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,4)), using
- linear OA(41, 5, F4, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([251,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([251,12]) [i] based on
(37, 54, 12217)-Net in Base 4 — Upper bound on s
There is no (37, 54, 12218)-net in base 4, because
- 1 times m-reduction [i] would yield (37, 53, 12218)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 81 134070 340263 332204 451599 903875 > 453 [i]