Routings In Optical Networks And Constant-Composition Codes

There are two parts in this thesis,we investigate the routings in optical networks in the first part and the constant-composition code in the second part.In the first part of this thesis we construct sets of routings in the complete directed graph(?)_n that tolerate up to f failures of nodes or links.Fault-tolerant routings with minimum optical index has been a topic of considerable recent interest.In optical networks,the vast bandwidth available in an optical fiber is utilized by partitioning it into several channels,each routing path uses a particular wavelength,and two paths can use the same wavelength if and only if they have no links in common.The optical networks relevant to our work use wavelength division multiplexing:these are called WDM optical networks.The notion of routings in optical networks was first introduced by Alok Aggarwal,Amotz Bar- Noy^?,Don Coppersmith,Rajiv Ramaswami,Raruch Schieber,Madhu Sudan.In general,we want to minimize the total number of wavelengths used in the network.It is a hard problem and only a few results have been found by now.In this thesis,we mainly use the methods in combinatorial designs to study routings in optical networks when n＞9.The main results,which we have got,are the following three points.1.We use a standard backtracking algorithm to find the solutions n=9,10.2.We give a method that construct large sets of disjoint Steiner triple systems from[46].This method plays an important role in constructing large sets of disjoint Steiner triple systems.We believe it will work on our problem.And we try to use this method to solve our problem completely.3.We carried out a pilot study,and have found some useful results about HLSTSs. And we get the necessary condition to construct the EGDDs.The above three points are the main results in the first part of this thesis.I think some results may be improved.For example,due to the algorithm and structure we selected,we can't prove the existence of EGDD(2,3,h³) when h is even.But this does not mean it does't exist.We can improve our algorithm and make it much more efficient or we can take other delicate structure to prove the existence of some special EPCS.In the second part of this thesis we investigate the constant-composition codes. A constant-composition code is a special constant-weight code under the restriction that each symbol should appear a given number of times in each codeword.Constantweight codes plays an important role in coding theory.Binary constant-weight codes have been extensively studied by many authors.The class of constant-composition codes includes the important permutation codes and has attracted recent interest due to their numerous applications.In general,we want to do some research on (n,5,[3,1])₃-codes.In this thesis,we mainly use the methods in combinatorial designs to study constant-composition codes.The main results,which we have got,are the following three points.1.We use some algorithm to construct some optimal constant-composition codes directly. 2.We use a method that construct from GDDs and GDCs,and we have constructed some useful results about GDCs.3.We construct optimal(24t+ 1,5,[3,1])3-Codes,t≥1.And we have proved that the(6,5,[3,1])₃-Codes of size 4 does not exist.The above three points are the main results in the second part of this thesis.I think some results may be improved.For example,due to the algorithm and structure we selected,we can't prove the existence of some optimal(n,5,[3,1])₃-Codes and the nonexistence of(9,5,[3,1])₃-Codes of size 12.We can improve our algorithm and make it much more efficient or we can take other delicate structure to prove the existence of some special GDCs and constant-composition codes.