Tato diplomová práce se věnuje problematice VLSI návrhu a implementaci vrstvy RS-FEC pro 400 Gb/s Ethernet do FPGA Intel® Stratix® 10 DX 2100. V práci je charakterizován současný stav rychlostí Ethernetu, význam a kontext samoopravných kódů v rámci protokolu Ethernet. Dále je popsána výroba PLD čipů i matematická podstata RS sa moopravných kódů. V části praktické je představen návrh řešení systému RS-FEC, který byl realizován genericky pomocí jazyka VHDL. Zároveň byly jeho komponenty implementovány a v závěrečné diskusi je popsáno jeho řešení, dosažené výsledky včetně jeho budoucího rozšíření.

Reed-Solomon codes are a well known family of error-correcting codes with many good properties. However, they require a finite field to operate, limiting the alphabet size to a prime power. In this work, we build a weaker algebraic structure which supports alphabet of any integer size and requires only standard addition, multiplication and division to implement. Then we study a family of error-correcting codes based on matrix multiplication over this structure. We also adapt the Reed-Solomon code principle on this code family and study its properties. We prove and verify experimentally that while a random code of this family has high distance, the Reed-Solomon adaptation fails to perform well. 1

Title: Goppa codes and their applications Author: Bc. Jaroslav Kotil Department: Department of algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc. Abstract: In this diploma paper we introduce Goppa codes, describe their para- metres and inclusion in Alternant codes, which are residual Generalized Reed- Solomon codes, and Algebraic-geometry codes. Aftewards we demonstrate deco- ding of Goppa codes and introduce Wild Goppa codes. We also describe post- quantum cryptography member: McEliece cryptosystem for which no effective attacks with quantum computers are known. We outline a usage of this crypto- system with Goppa codes and describe the security of the cryptosystem together with possible attacks of which the most effective ones are based on information- set decoding. Keywords: Goppa codes, Generalized Reed-Solomon codes, Algebraic-geometry codes, Post-quantum cryptography, McEliece cryptosystem 1

The main objective of this thesis was to create a model of forward error correction (FEC) system used in ADSL modem, which will be able to check its individual settings using the test methods specified in Recommendation G.996.1. The work is divided into two basic parts – theoretical and practical. The first part describes the basic functional characteristics of ADSL system, with attention being paid to aspects affecting the activity and setting of the error protection system. There are also discussed the theoretical basics of FEC coding techniques used in ADSL. The work also contains a description of the method for testing DSL system resistance against impulsive interference, which is specified in Recommendation G.996.1. The practical part of this thesis contains a description of the proposed model of the FEC system. The model was implemented in the programming environment Matlab Simulink. The proposed model includes the simulation of FEC coding process, DMT modulation and transmission channel with the impulse interference and the background noise. The proposed program is able to calculate parameters for setting FEC system from the specified data transfer rates and the requirement for interleaving. In addition, allows you to set the parameters of transmission line and sources of impulse interference and of the background noise. Under the set of input characteristics of the transmission is calculated bit allocation and the resulting parameters are given to a computational core of the model. In the application can be performed two types of test tasks. The first one is based on the Recommendation G.996.1 and makes it possible to determine probability of errored second. The second test task is for the measurement of resistance of the transmission system against impulsive interference. The results of the simulations are summarized in the conclusion of work.

The doctoral thesis deals with design of coding for frequency hopping stations in band with intensive jamming. In digital modulations erroneous determination of the modulation state occurs due to jam at the receiver side. The result is erroneously transferred symbols of the message. Errors created during the transmission can be eliminated by using error control systems. It is also possible to prevent these errors by using algorithms (techniques) of frequency hopping which select the appropriate channel. Appropriate communication channel is a channel with a lower probability of erroneous symbol in the message. The main contribution of this thesis is to design a new frequency hopping technique with collision avoidance (FH/CA). The station with FH/CA technique measures signal levels in the considered several channels before every jump. Based on the measurements the most appropriate channel with the lowest value of measured signal level is selected. Therefore, it is more probable that a jump to an unoccupied channel with a transmission will occur. Using a mathematical model, the performance of the newly proposed FH/CA technique is compared with the currently used techniques FH and AFH. Comparison criteria are the probability of a collision between an FH/CA communication system and a static (device transmitting continuously at a fixed frequency) or dynamic jammer (i.e. other FH or AFH systems). By comparing the values of the probability of jammed transmission, indisputable theoretical advantages of the new FH/CA technique were found, compared to the currently used FH and AFH techniques. The FH/CA technique always has better or equal results compared with the FH technique in the case of interference by static and dynamic jammers. The FH/CA technique in a band with static and dynamic jammers usually has better results than the AFH technique. A significant contribution of the FH/CA technique can be seen in the case of dynamic jammers. On the other hand, in the case of static jammers the FH/CA technique is in certain situations worse than the AFH technique. The accuracy of the mathematical models were successfully verified on a simulation model that was created as a part of this thesis in the MATLAB environment. Based on the obtained data from the model there was designed coding for frequency hopping stations with the new technique of frequency hopping FH/CA which is designed for small-volume data transfer in a band with intensive jamming.