Best Known (83, 112, s)-Nets in Base 4
(83, 112, 531)-Net over F4 — Constructive and digital
Digital (83, 112, 531)-net over F4, using
- 2 times m-reduction [i] based on digital (83, 114, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 38, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 38, 177)-net over F64, using
(83, 112, 1055)-Net over F4 — Digital
Digital (83, 112, 1055)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4112, 1055, F4, 29) (dual of [1055, 943, 30]-code), using
- 16 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 8 times 0) [i] based on linear OA(4108, 1035, F4, 29) (dual of [1035, 927, 30]-code), using
- construction XX applied to C1 = C([319,345]), C2 = C([317,343]), C3 = C1 + C2 = C([319,343]), and C∩ = C1 ∩ C2 = C([317,345]) [i] based on
- linear OA(4101, 1023, F4, 27) (dual of [1023, 922, 28]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {319,320,…,345}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(4101, 1023, F4, 27) (dual of [1023, 922, 28]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {317,318,…,343}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(4106, 1023, F4, 29) (dual of [1023, 917, 30]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {317,318,…,345}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(496, 1023, F4, 25) (dual of [1023, 927, 26]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {319,320,…,343}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(41, 6, F4, 1) (dual of [6, 5, 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
- linear OA(41, 6, F4, 1) (dual of [6, 5, 2]-code) (see above)
- construction XX applied to C1 = C([319,345]), C2 = C([317,343]), C3 = C1 + C2 = C([319,343]), and C∩ = C1 ∩ C2 = C([317,345]) [i] based on
- 16 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 8 times 0) [i] based on linear OA(4108, 1035, F4, 29) (dual of [1035, 927, 30]-code), using
(83, 112, 119611)-Net in Base 4 — Upper bound on s
There is no (83, 112, 119612)-net in base 4, because
- 1 times m-reduction [i] would yield (83, 111, 119612)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 6 740673 967035 956119 465042 230310 631568 499453 904949 686732 941151 145412 > 4111 [i]