Best Known (53−15, 53, s)-Nets in Base 4
(53−15, 53, 240)-Net over F4 — Constructive and digital
Digital (38, 53, 240)-net over F4, using
- t-expansion [i] based on digital (37, 53, 240)-net over F4, using
- 1 times m-reduction [i] based on 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
- trace code for nets [i] based on digital (1, 18, 80)-net over F64, using
- 1 times m-reduction [i] based on digital (37, 54, 240)-net over F4, using
(53−15, 53, 391)-Net over F4 — Digital
Digital (38, 53, 391)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(453, 391, F4, 15) (dual of [391, 338, 16]-code), using
- 120 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0, 1, 16 times 0, 1, 22 times 0, 1, 28 times 0, 1, 33 times 0) [i] based on linear OA(445, 263, F4, 15) (dual of [263, 218, 16]-code), using
- construction XX applied to C1 = C([254,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([254,13]) [i] based on
- linear OA(441, 255, F4, 14) (dual of [255, 214, 15]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(441, 255, F4, 14) (dual of [255, 214, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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 = {−1,0,…,13}, 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(40, 4, F4, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(40, 4, F4, 0) (dual of [4, 4, 1]-code) (see above)
- construction XX applied to C1 = C([254,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([254,13]) [i] based on
- 120 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0, 1, 16 times 0, 1, 22 times 0, 1, 28 times 0, 1, 33 times 0) [i] based on linear OA(445, 263, F4, 15) (dual of [263, 218, 16]-code), using
(53−15, 53, 33432)-Net in Base 4 — Upper bound on s
There is no (38, 53, 33433)-net in base 4, because
- 1 times m-reduction [i] would yield (38, 52, 33433)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 20 282924 551218 469073 070087 311896 > 452 [i]