November  2016, 10(4): 847-850. doi: 10.3934/amc.2016044

Some new two-weight ternary and quinary codes of lengths six and twelve

1. 

University of Central Oklahoma, 100 North University Drive, P.O. Box 129, Edmond, OK 73034, United States

2. 

Colorado State University, 841 Oval Drive, P.O. Box 1874, Fort Collins, CO 80523-1874, United States

Received  February 2015 Published  November 2016

Let $[n,k]_q$ be a projective two-weight linear code over ${\rm GF}(q)^n$. In this correspondence, 9 codes are constructed in which $k=3$.
Citation: Liz Lane-Harvard, Tim Penttila. Some new two-weight ternary and quinary codes of lengths six and twelve. Advances in Mathematics of Communications, 2016, 10 (4) : 847-850. doi: 10.3934/amc.2016044
References:
[1]

A. S. Barlotti, $\{ k;n\}$-archi di un piano lineare finite, Boll. Un. Mat. Ital., 11 (1956), 553-556.  Google Scholar

[2]

L. M. Batten and J. M. Dover, Some sets of type $(m,n)$ in cubic order planes, Des. Codes Cryptogr., 16 (1999), 211-213. doi: 10.1023/A:1008397209409.  Google Scholar

[3]

A. Blokhuis and M. Lavrauw, Scattered spaces with respect to a spread in $PG(n,q)$, Geom. Dedicata, 81 (2000), 231-243. doi: 10.1023/A:1005283806897.  Google Scholar

[4]

R. C. Bose, Mathematical theory of the symmetrical factorial design, Sankhyā Indian J. Stat., 8 (1947), 107-166.  Google Scholar

[5]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I. The user language, J. Symb. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.  Google Scholar

[6]

R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122. doi: 10.1112/blms/18.2.97.  Google Scholar

[7]

L. R. A Casse, W. A. Jackson, T. Penttila and G. F. Royle, Sets of type $(m,n)$ in $PG(2, r^2)$, $r$ odd,, in preparation., ().   Google Scholar

[8]

W. Cherowitzo, $\alpha$-flocks and hyperovals, Geom. Dedicata, 72 (1998), 221-246. doi: 10.1023/A:1005022808718.  Google Scholar

[9]

W. Cherowitzo, C. M. O'Keefe and T. Penttila, A unified construction of finite geometries associated with q-clans in characteristic 2, Adv. Geom., 3 (2003), 1-21. doi: 10.1515/advg.2003.002.  Google Scholar

[10]

W. Cherowitzo, T. Penttila, I. Pinneri and G. F. Royle, Flocks and ovals, Geom. Dedicata, 60 (1996), 17-37. doi: 10.1007/BF00150865.  Google Scholar

[11]

F. De Clerck, S. De Winter and T. Maes, A geometric approach to Mathon maximal arcs, J. Combin. Theorey Ser. A, 118 (2001), 1196-1211. doi: 10.1016/j.jcta.2010.12.004.  Google Scholar

[12]

F. De Clerck, S. De Winter and T. Maes, Partial flocks of the quadratic cone yielding Mathon maximal arcs, Discrete Math., 312 (2012), 2421-2428. doi: 10.1016/j.disc.2012.04.028.  Google Scholar

[13]

F. De Clerck, S. De Winter and T. Maes, Singer 8-arcs of Mathon type in $PG(2,2^7)$, Des. Codes Cryptogr., 64 (2012), 17-31. doi: 10.1007/s10623-011-9502-4.  Google Scholar

[14]

M. J. de Resmini, A 35-set of type $(2,5)$ in $PG(2,9)$, J. Combin. Theory Ser. A, 45 (1987), 303-305. doi: 10.1016/0097-3165(87)90021-5.  Google Scholar

[15]

M. J. de Resmini and G. Migliori, A 78-set of type $(2,6)$ in $PG(2,16)$, Ars Combin., 22 (1986), 73-75.  Google Scholar

[16]

The GAP Group, GAP - Groups, Algorithms, and Programming,, available online at , ().   Google Scholar

[17]

N. Hamilton, Maximal Arcs in Finite Projective Planes and Associated Structure in Projective Spaces, Ph.D. thesis, Univ. Western Australia, 1995. Google Scholar

[18]

N. Hamilton, Degree 8 maximal arcs in $PG(2,2^h)$, $h$ odd, J. Combin. Theory Ser. A, 100 (2002), 265-276. doi: 10.1006/jcta.2002.3297.  Google Scholar

[19]

N. Hamilton and R. Mathon, More maximal arcs in Desarguesian projective planes and their geometric structure, Adv. Geom., 3 (2003), 251-261. doi: 10.1515/advg.2003.015.  Google Scholar

[20]

N. Hamilton and R. Mathon, On the spectrum of non-Denniston maximal arcs in $PG(2,2h)$, Europ. J. Combin., 25 (2004), 415-421. doi: 10.1016/j.ejc.2003.07.006.  Google Scholar

[21]

N. Hamilton and T. Penttila, Sets of type $(a,b)$ from subgroups of $\GammaL(1,p^R)$, J. Algebr. Combin., 13 (2001), 67-76. doi: 10.1023/A:1008775818040.  Google Scholar

[22]

R. Mathon, New maximal arcs in Desarguesian planes, J. Combin. Theory Ser. A, 97 (2002), 353-368. doi: 10.1006/jcta.2001.3218.  Google Scholar

[23]

C. M. O'Keefe and T. Penttila, A new hyperoval in $PG(2,32)$, J. Geom., 44 (1992), 117-139. doi: 10.1007/BF01228288.  Google Scholar

[24]

S. E. Payne, A new infinite family of generalized quadrangles, Congr. Numer., 49 (1985), 115-128.  Google Scholar

[25]

T. Penttila and G. F. Royle, Sets of type $(m,n)$ in the affine and projective planes of order nine, Des. Codes Cryptogr., 6 (1995), 229-245. doi: 10.1007/BF01388477.  Google Scholar

[26]

B. Schmidt and C. White, All two-weight irreducible cyclic codes?, Finite Fields Appl., 8 (2002), 1-17. doi: 10.1006/ffta.2000.0293.  Google Scholar

[27]

B. Segre, Forme e geometrie hermitiane, con particolare riguardo al caso finite, Ann. Mat. Pura Appl., 70 (1965), 1-201.  Google Scholar

[28]

M. Tallini Scafati, $\{k,n\}$-archi di un pinao grafico finito, con particolare riguardo a quelli con due caratteri. I, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 40 (1966), 812-818.  Google Scholar

[29]

M. Tallini Scafati, $\{k,n\}$-archi di un pinao grafico finito, con particolare riguardo a quelli con due caratteri. II, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 40 (1966), 1020-1025.  Google Scholar

show all references

References:
[1]

A. S. Barlotti, $\{ k;n\}$-archi di un piano lineare finite, Boll. Un. Mat. Ital., 11 (1956), 553-556.  Google Scholar

[2]

L. M. Batten and J. M. Dover, Some sets of type $(m,n)$ in cubic order planes, Des. Codes Cryptogr., 16 (1999), 211-213. doi: 10.1023/A:1008397209409.  Google Scholar

[3]

A. Blokhuis and M. Lavrauw, Scattered spaces with respect to a spread in $PG(n,q)$, Geom. Dedicata, 81 (2000), 231-243. doi: 10.1023/A:1005283806897.  Google Scholar

[4]

R. C. Bose, Mathematical theory of the symmetrical factorial design, Sankhyā Indian J. Stat., 8 (1947), 107-166.  Google Scholar

[5]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I. The user language, J. Symb. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.  Google Scholar

[6]

R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122. doi: 10.1112/blms/18.2.97.  Google Scholar

[7]

L. R. A Casse, W. A. Jackson, T. Penttila and G. F. Royle, Sets of type $(m,n)$ in $PG(2, r^2)$, $r$ odd,, in preparation., ().   Google Scholar

[8]

W. Cherowitzo, $\alpha$-flocks and hyperovals, Geom. Dedicata, 72 (1998), 221-246. doi: 10.1023/A:1005022808718.  Google Scholar

[9]

W. Cherowitzo, C. M. O'Keefe and T. Penttila, A unified construction of finite geometries associated with q-clans in characteristic 2, Adv. Geom., 3 (2003), 1-21. doi: 10.1515/advg.2003.002.  Google Scholar

[10]

W. Cherowitzo, T. Penttila, I. Pinneri and G. F. Royle, Flocks and ovals, Geom. Dedicata, 60 (1996), 17-37. doi: 10.1007/BF00150865.  Google Scholar

[11]

F. De Clerck, S. De Winter and T. Maes, A geometric approach to Mathon maximal arcs, J. Combin. Theorey Ser. A, 118 (2001), 1196-1211. doi: 10.1016/j.jcta.2010.12.004.  Google Scholar

[12]

F. De Clerck, S. De Winter and T. Maes, Partial flocks of the quadratic cone yielding Mathon maximal arcs, Discrete Math., 312 (2012), 2421-2428. doi: 10.1016/j.disc.2012.04.028.  Google Scholar

[13]

F. De Clerck, S. De Winter and T. Maes, Singer 8-arcs of Mathon type in $PG(2,2^7)$, Des. Codes Cryptogr., 64 (2012), 17-31. doi: 10.1007/s10623-011-9502-4.  Google Scholar

[14]

M. J. de Resmini, A 35-set of type $(2,5)$ in $PG(2,9)$, J. Combin. Theory Ser. A, 45 (1987), 303-305. doi: 10.1016/0097-3165(87)90021-5.  Google Scholar

[15]

M. J. de Resmini and G. Migliori, A 78-set of type $(2,6)$ in $PG(2,16)$, Ars Combin., 22 (1986), 73-75.  Google Scholar

[16]

The GAP Group, GAP - Groups, Algorithms, and Programming,, available online at , ().   Google Scholar

[17]

N. Hamilton, Maximal Arcs in Finite Projective Planes and Associated Structure in Projective Spaces, Ph.D. thesis, Univ. Western Australia, 1995. Google Scholar

[18]

N. Hamilton, Degree 8 maximal arcs in $PG(2,2^h)$, $h$ odd, J. Combin. Theory Ser. A, 100 (2002), 265-276. doi: 10.1006/jcta.2002.3297.  Google Scholar

[19]

N. Hamilton and R. Mathon, More maximal arcs in Desarguesian projective planes and their geometric structure, Adv. Geom., 3 (2003), 251-261. doi: 10.1515/advg.2003.015.  Google Scholar

[20]

N. Hamilton and R. Mathon, On the spectrum of non-Denniston maximal arcs in $PG(2,2h)$, Europ. J. Combin., 25 (2004), 415-421. doi: 10.1016/j.ejc.2003.07.006.  Google Scholar

[21]

N. Hamilton and T. Penttila, Sets of type $(a,b)$ from subgroups of $\GammaL(1,p^R)$, J. Algebr. Combin., 13 (2001), 67-76. doi: 10.1023/A:1008775818040.  Google Scholar

[22]

R. Mathon, New maximal arcs in Desarguesian planes, J. Combin. Theory Ser. A, 97 (2002), 353-368. doi: 10.1006/jcta.2001.3218.  Google Scholar

[23]

C. M. O'Keefe and T. Penttila, A new hyperoval in $PG(2,32)$, J. Geom., 44 (1992), 117-139. doi: 10.1007/BF01228288.  Google Scholar

[24]

S. E. Payne, A new infinite family of generalized quadrangles, Congr. Numer., 49 (1985), 115-128.  Google Scholar

[25]

T. Penttila and G. F. Royle, Sets of type $(m,n)$ in the affine and projective planes of order nine, Des. Codes Cryptogr., 6 (1995), 229-245. doi: 10.1007/BF01388477.  Google Scholar

[26]

B. Schmidt and C. White, All two-weight irreducible cyclic codes?, Finite Fields Appl., 8 (2002), 1-17. doi: 10.1006/ffta.2000.0293.  Google Scholar

[27]

B. Segre, Forme e geometrie hermitiane, con particolare riguardo al caso finite, Ann. Mat. Pura Appl., 70 (1965), 1-201.  Google Scholar

[28]

M. Tallini Scafati, $\{k,n\}$-archi di un pinao grafico finito, con particolare riguardo a quelli con due caratteri. I, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 40 (1966), 812-818.  Google Scholar

[29]

M. Tallini Scafati, $\{k,n\}$-archi di un pinao grafico finito, con particolare riguardo a quelli con due caratteri. II, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 40 (1966), 1020-1025.  Google Scholar

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