Math 525 — Algebraic Coding Theory — Syllabus

Linear codes, perfect and related codes, cyclic linear codes, BCH codes, burst error-correcting codes.

1. Math 254 (Linear Algebra)

This course is intended for advanced undergraduate students and graduate students at any level. The fundamentals of coding theory and the most widely used error-correcting codes are studied. Although the theory of error-correcting codes started in electrical engineering, it soon became a mathematical topic. In a seminal paper published in 1948, Claude Shannon demonstrated that by properly encoding information (i.e., by adding some redundancy to it), errors introduced by a noisy channel or storage medium could be reduced to any desired level without sacrificing the rate of information transmission or storage. Although Shannon’s result was a fundamental step, he did not indicate how to effectively construct the codes promised by his theory. This marked the birth of coding theory. The design of the so-called good codes involves techniques from algebra (linear and abstract), number theory, combinatorics, and algebraic geometry. Hence, results in the theory of ECC can be regarded as applications of those areas of mathematics. From the practical point of view, error control codes provide the means for achieving the high degree of efficiency and reliability required in modern data transmission and storage systems. The control of errors that occur during the transmission of information is a major concern of the designer of such systems. The use of coding for error control is an integral part in the design of communication systems.

• Recognize the practical need for ECC and explain how they work.
• Calculate theoretical limits on error-correction.
• Describe the relevance of linear codes and explain how they are encoded and decoded.
• Calculate the minimum distance of a linear code from one of its defining matrices.
• Describe the relevance of cyclic codes and explain how they are encoded.
• Construct a finite field and perform operations (addition, multiplication, and division) with its elements.
• Construct a BCH code meeting certain requirements on rate and error-correction capability.
• Construct Reed-Solomon codes (used for error-correction in CDs, DVDs, Blu-ray discs), and describe their general decoding principle.
• Design a code that will correct bursts of errors of a given length.

If you are a student with a disability and believe you will need accommodations for this class, it is your responsibility to contact the Student Ability Success Center at (619) 594-6473. To avoid any delay in the receipt of your accommodations, you should contact Student Ability Success Center as soon as possible. Please note that accommodations are not retroactive, and that I cannot provide accommodations based upon disability until I have received an accommodation letter from Student Ability Success Center. Your cooperation is appreciated.

The Family Educational Rights and Privacy Act (FERPA) mandates the protection of student information, including contact information, grades, and graded assignments. I will not post grades or leave graded assignments in public places. Students will be notified at the time of an assignment if copies of student work will be retained beyond the end of the semester or used as examples for future students or the wider public. Students maintain intellectual property rights to work products they create as part of this course unless they are formally notified otherwise.

The SDSU Math & Stat Learning Center is in the Love Library, Room LL-328. "The Math and Stats Learning Center is open to support students in all lower division math courses at SDSU. We have tutors available for walk-in help during all open hours. TAs for Math 141, 150, 151, and 252 also hold their office hours there. Please see the schedule of when the TAs for your class will be in the center by going to our website: mlc.sdsu.edu. The MLC is supported by your student success fee. We strongly encourage you to use this wonderful, free resource. Some students believe that they should not need to ask for help. But, research has shown that the average grade for students who attend the MLC is one half grade higher than those who don’t seek such support."

If you are enrolled in a class which does not have targeted support, the MLC can still serve as a great math study/meeting place; and if you are interested in becoming a tutor in the center, keep an eye on the center's webpage for hiring announcements.

Students are generally encouraged to study together, and to work together to solve exercises. Finals, Midterms, Quizzes, Project, and other designated "individual work" activities must be completed without assistance. All violations will be reported to the Center for Student Rights and Responsibilities and will also result in score/grade reductions at the professor's discretion. Please review SDSU's full policy on academic honesty.

Examples of academic dishonesty include but are not limited to

• copying, in part or in whole, from another's test or other examination
• obtaining copies of a test, an examination, or other course material without the permission of the instructor
• collaborating with another or others in work to be presented without the permission of the instructor
• falsifying records, laboratory work, or other course data
• submitting work previously presented in another course, if contrary to the rules of the course
• altering or interfering with grading procedures
• assisting another student in any of the above
• using sources verbatim or paraphrasing without giving proper attribution (this can include phrases, sentences, paragraphs and/or pages of work)
• copying and pasting work from an online or offline source directly and calling it your own
• using information you find from an online or offline source without giving the author credit
• replacing words or phrases from another source and inserting your own words or phrases

According to the University Policy File, students should notify the instructors of affected courses of planned absences for religious observances by the end of the second week of classes.

Students are instructed to contact their professor/instructor in the event they need to miss class, etc. due to an illness, injury or emergency. All decisions about the impact of an absence, as well as any arrangements for making up work, rest with the instructors. Student Health Services (SHS) does not provide medical excuses for short-term absences due to illness or injury. When a medical-related absence persists beyond five days, SHS will work with students to provide appropriate documentation. When a student is hospitalized or has a serious, ongoing illness or injury, SHS will, at the student's request and with the student’s consent, communicate with the student’s instructors via the Vice President for Student Affairs and may communicate with the student’s Assistant Dean and/or the Student Ability Success Center

All SDSU instructors are mandated reporters, and are thus required to share information regarding sexual violence on SDSU's campus with the Title IX Coordinator, Jessica Rentto 619-594-6017. The Title IX Coordinator has access to accommodations and support services at SDSU, and possibilities for holding accountable the person who harmed you. You are not required to share information you do not wish to disclose and your level of involvement will be at your discretion. If you do not want the Title IX Officer notified, instead of disclosing this information to your instructor, you can speak confidentially with the following people on campus and in the community: Sexual Violence Victim Advocate 619-594-0210 or Counseling and Psychological Services 619-594-5220, [email protected]. They can connect you with support services and discuss options for pursuing a University or criminal investigation. For more information regarding your university rights and options as a survivor of sexual misconduct or sexual violence, please visit titleix.sdsu.edu.

For millennia, the Kumeyaay people have been a part of this land. This land has nourished, healed, protected and embraced them for many generations in a relationship of balance andharmony. As members of the San Diego State community we acknowledge this legacy. We promote this balance and harmony. We find inspiration from this land; the land of theKumeyaay.