| The purpose of communication is to transmit unknown information timely and reliably to the receiver. So, a communication system transmitting information must be reliable and rapid. But reliable and rapid often conflict each other, the theory of error-correcting code arose to solve this contradiction. A error correction code with the capability of detection or correction is the art of adding redundancy efficiently so that most messages, if distorted, can be correctly decodes. In the first21st century, Alber et al. introduced a special quantum error-correcting code called quantum jump code which correct errors caused by jumps in order to reduce the redundancy.In2003, Beth et al. introduced a combination configuration called a spontaneous emission error design (SEED) contacting with quantum jump code:let n, k, t, m be integers with0<t<k<n and V be a set of n elements. A t-(n, k; m)-SEED is a system B of k-subsets of V with a partition B1, B2,...,Bm of B satisfying that|{B∈S:E(?) B}|/|Bi|==λE, for any1≤1≤m and E (?) V,|E|≤<t, where AE is a constant depending only on E. In addition, they also proved that a spontaneous emission error design is a quantum jump code with the same parameters.One main aim of this article is to discuss in detail the relation between the quantum jump codes and spontaneous emission error designs and prove the equivalence of ex-istence between some quantum jump codes with specified parameters and spontaneous emission error designs. Moreover, by using the equivalence and the nonexistence of a spontaneous emission error design with specified parameters to prove the nonexistence of some quantum jump codes.In1948, Shannon published the paper "A Mathematical Theory of Communica-tion" and pointed out that:as long as using appropriately error-correction code, the message can be transmitted in a multi-channel. So the error-correcting code arose. Since1948, people try unremittingly to find many good codes in order to meet many practical requirements.Another aim of this article is to construct mutually disjoint t-designs from classic error-correcting good codes, such as extremal doubly-even self-dual codes and extremal ternary self-dual codes, in order that spontaneous emission error designs can be con-structed. Moreover, spontaneous emission error designs also be constructed from the octacode O8and the lifted Golay code G24over Z4. The main results are divided into four parts as follows corresponding to Chapter2,3,4,5in this article,respectively,and this is also the main work of my five papers when I’m a doctoral student.1.The nonexistence of some(n,m,t)k-JCs is proved,where m=(n-t k-t),k≥t+1, and(n,t,k)=(2k+1,1,k),(2k,2,k),(7,2,3),(8,3,4).2.The construction of3.(n,k;m)-SEEDs with(n,k,m)∈{(8,4,3),(32,8,5),(56,12,9),(56,16,9),(56,24,9),(80,16,52)} and5-(48,k；506)-SEEDs for k=12,16,20,24is given from extremal doubly-even self.dual codes.3.The construction of a list of3-and5.SEEDs is given from extremal ternary self-dual codes as follows.3-(16,k;28)-SEEDs for k=6,9;3-(28,k;26)-SEEDs for k=9,12,15;3-(40,k;6916)-SEEDs for k=12,15,18,21;3-(64,k;124)-SEEDs for k=18,21,24,27,30,33.5-(24,k;121)-SEEDs for k=9,15and a5-(24,12;66)-SEEDs;5-(48,k;1035)-SEEDs for k=15,18,21,27and a5-(48,24；529)-SEEDs;5-(36,k;34)-SEEDs for k=12,15,21and a5-(36,18;17)-SEEDs;5-(60,k;58)-SEEDs for k=18,21,24,27,33and a5-(60,30；29)-SEEDs.4.The construction of3-(8,k;3)-SEEDs for k=4,5and5-(24,k;22)-SEEDs for k=8,10,12,13is given from the octacode O8and the lifted Golay code G24over Z4, respectively. |
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