| lower bound: | 80 |
| upper bound: | 88 |
Construction of a linear code [109,9,80] over GF(8):
[1]: [1, 1, 1] Cyclic Linear Code over GF(2^3)
RepetitionCode of length 1
[2]: [108, 8, 80] Quasicyclic of degree 4 Linear Code over GF(2^3)
QuasiCyclicCode of length 108 with generating polynomials: x^19 + w*x^18 + w^6*x^17 + w^5*x^16 + w^5*x^14 + w*x^13 + w^3*x^12 + w^2*x^11 + w^6*x^10 + w^6*x^9 + w^2*x^8 + w^3*x^7 + w*x^6 + w^5*x^5 + w^5*x^3 + w^6*x^2 + w*x + 1, w^6*x^26 + w*x^24 + w^5*x^23 + w^5*x^22 + w^5*x^21 + w^3*x^20 + w^2*x^19 + w^6*x^18 + w*x^17 + w^3*x^16 + w^3*x^15 + w^6*x^14 + w^2*x^13 + w*x^12 + w^5*x^11 + w^4*x^9 + w^4*x^8 + w^4*x^7 + x^6 + w^5*x^5 + w*x^4 + w*x^3 + w^2, w^2*x^26 + x^23 + w^2*x^22 + w^3*x^21 + w^2*x^20 + w^2*x^18 + w^5*x^17 + w^5*x^16 + w^3*x^15 + x^14 + w^2*x^13 + w^4*x^12 + w^5*x^11 + w^5*x^10 + w^6*x^9 + w^4*x^7 + w*x^6 + w^3*x^5 + w^5*x^4 + w^3*x^3 + w^3*x^2 + x + w^4, x^26 + w^4*x^25 + w^3*x^24 + w^2*x^22 + w^4*x^21 + w^3*x^19 + x^18 + w^4*x^17 + w^3*x^16 + w^2*x^15 + w*x^14 + w^5*x^13 + w^4*x^12 + w^2*x^11 + x^10 + w^4*x^9 + w*x^8 + w^3*x^7 + w^6*x^6 + w^6*x^4 + w^6*x^3 + w^3*x^2 + w*x
[3]: [108, 9, 79] Quasicyclic of degree 4 Linear Code over GF(2^3)
QuasiCyclicCode of length 108 with generating polynomials: x^18 + w^3*x^17 + w^4*x^16 + x^15 + x^14 + w^4*x^13 + w^2*x^12 + w^5*x^11 + w^3*x^10 + w^4*x^9 + w^3*x^8 + w^5*x^7 + w^2*x^6 + w^4*x^5 + x^4 + x^3 + w^4*x^2 + w^3*x + 1, w^5*x^26 + w*x^25 + w*x^24 + w^5*x^22 + w^5*x^20 + w^2*x^19 + w^6*x^17 + w^5*x^16 + w^2*x^15 + w^5*x^14 + w*x^13 + w^4*x^12 + w^2*x^11 + w^3*x^10 + w^3*x^9 + w^6*x^8 + w^3*x^7 + w^6*x^6 + w^2*x^5 + w^3*x^4 + x^3 + w^3*x^2 + w^3*x + w^3, w^6*x^26 + x^25 + x^24 + x^23 + w^2*x^21 + w^5*x^20 + w^3*x^19 + w^3*x^18 + w^5*x^17 + w^5*x^15 + w^2*x^14 + w^6*x^13 + x^12 + w^5*x^11 + w^5*x^9 + w*x^8 + w*x^7 + w^2*x^6 + w^4*x^5 + w^6*x^4 + w*x^3 + x^2 + w*x + w^3, w^6*x^26 + w^2*x^25 + w*x^24 + x^23 + x^22 + w^6*x^21 + w^3*x^20 + w^3*x^19 + x^17 + w^5*x^16 + w^2*x^15 + w*x^13 + w^6*x^12 + w^3*x^11 + w^5*x^10 + w^4*x^9 + w*x^7 + x^6 + w^2*x^5 + w^2*x^4 + x^3 + w^2*x^2 + w^5*x + w^6
[4]: [109, 9, 80] Linear Code over GF(2^3)
ConstructionX using [3] [2] and [1]
last modified: 2026-09-29
Lb(109,9) = 79 is found by shortening of: Lb(110,10) = 79 is found by truncation of: Lb(117,10) = 86 DaH Ub(109,9) = 88 follows by a one-step Griesmer bound from: Ub(20,8) = 11 is found by considering shortening to: Ub(17,5) = 11 is found by considering truncation to: Ub(16,5) = 10 MPa
MPa: Stefano Marcugini, Alfredo Milani & Fernanda Pambianco, email 2000-2002. Idem, Existence and classification of NMDS codes over GF(5) and GF(7), Seventh International Workshop on Alg. and Combin. Coding Theory, Bansko, Bulgaria, 18-24 June 2000. Idem, NMDS Codes of Maximal Length Over F_q, 8<=q<=11, IEEE Transactions on Informations Theory, 48, No.4, 2002.
Notes
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