Best Known (122−31, 122, s)-Nets in Base 4
(122−31, 122, 531)-Net over F4 — Constructive and digital
Digital (91, 122, 531)-net over F4, using
- 4 times m-reduction [i] based on digital (91, 126, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 42, 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, 42, 177)-net over F64, using
(122−31, 122, 576)-Net in Base 4 — Constructive
(91, 122, 576)-net in base 4, using
- 1 times m-reduction [i] based on (91, 123, 576)-net in base 4, using
- trace code for nets [i] based on (9, 41, 192)-net in base 64, using
- 1 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
- base change [i] based on digital (3, 36, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 36, 192)-net over F128, using
- 1 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
- trace code for nets [i] based on (9, 41, 192)-net in base 64, using
(122−31, 122, 1147)-Net over F4 — Digital
Digital (91, 122, 1147)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4122, 1147, F4, 31) (dual of [1147, 1025, 32]-code), using
- 108 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 8 times 0, 1, 19 times 0, 1, 32 times 0, 1, 42 times 0) [i] based on linear OA(4116, 1033, F4, 31) (dual of [1033, 917, 32]-code), using
- construction XX applied to C1 = C([1022,28]), C2 = C([0,29]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([1022,29]) [i] based on
- linear OA(4111, 1023, F4, 30) (dual of [1023, 912, 31]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,28}, and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4111, 1023, F4, 30) (dual of [1023, 912, 31]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,29], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4116, 1023, F4, 31) (dual of [1023, 907, 32]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,29}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(4106, 1023, F4, 29) (dual of [1023, 917, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 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, 5, F4, 0) (dual of [5, 5, 1]-code) (see above)
- construction XX applied to C1 = C([1022,28]), C2 = C([0,29]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([1022,29]) [i] based on
- 108 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 8 times 0, 1, 19 times 0, 1, 32 times 0, 1, 42 times 0) [i] based on linear OA(4116, 1033, F4, 31) (dual of [1033, 917, 32]-code), using
(122−31, 122, 153896)-Net in Base 4 — Upper bound on s
There is no (91, 122, 153897)-net in base 4, because
- 1 times m-reduction [i] would yield (91, 121, 153897)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 7 067721 706028 560622 648300 422204 716196 637793 031787 610742 969499 493272 261536 > 4121 [i]