In general, LDPC codes are classified into two categories based on their construction methods: algebraic methods and graphical methods. LDPC codes constructed based on finite geometries and finite fields are classified as algebraic LDPC codes, such as the cyclic and quasi-cyclic LDPC codes presented into .

Cyclic codes form an important subclass of linear block codes. These codes are attractive for two reasons: first, encoding and syndrome computation can be implemented easily by using simple shift-registers with linear feedback connections, namely, linear feedback shift-registers (LFSRs); and second, because they have considerable inherent algebraic structure, it is possible to devise various practical algorithms for decoding them. Cyclic codes have been widely used in communication and storage systems for error control. They are particularly efficient for error detection.

Finite fields have been applied to construct error-correcting codes for reliable information transmission and data storage [–] since the late 1950s. These codes are commonly called algebraic codes, which have nice structures and large minimum distances. The most well-known classical algebraic codes are BCH and RS codes presented inandwhich can be decoded with the elegant hard-decision Berlekamp–Massey iterative algorithm.

Just hours after Democratic presidential nominee Joe Biden announced US Senator Kamala Harris of California as his vice-presidential running mate on August 11, 2020, President Donald Trump reacted at a White House press briefing, repeatedly calling her “nasty” and “horrible.”

Polar codes, discovered by Arikan [] in 2009, form a class of codes which provably achieve the capacity for a wide range of channels. Construction, encoding, and decoding of these codes are based on the phenomenon of channel polarization. In this chapter, we introduce polar codes from an algebraic point of view.

Apart from the construction of BCH, RS, finite-geometry, and RM codes based on finite fields and finite geometries, there are other methods (or techniques) for constructing long powerful codes from good short codes.

President Joe Biden gave his first address to Congress on April 28, 2021. He stood at the center of the US House rostrum, with two women seated behind him: Vice President Kamala Harris and House Speaker Nancy Pelosi. Biden began his speech by recognizing the historic moment of which he was part: “Madam Speaker, Madam Vice President. No president has ever said those words from this podium.” He added, “And it’s about time.” Biden was right. Never before 2021 had a woman served as vice president (and thus president of the Senate), nor had the first and only woman Speaker of the House – Nancy Pelosi – sat next to another woman during a presidential address to Congress. But it is the full picture of that moment, including Biden’s position at the podium as yet another white, male president of the United States, that captures the complexities of gender in the US elections in the early decades of the twenty-first century.

Besides Euclidean and projective geometries, there are other types of finite geometries. One such type is known as partial geometries. Akin to Euclidean and projective geometries, partial geometries can be used to construct LDPC codes whose Tanner graphs have similar structural properties as those of Euclidean- and projective-geometry LDPC codes.

Women voters have received special attention from the presidential candidates in recent elections primarily because of differences between women and men in their political preferences, a phenomenon commonly referred to as the gender gap. Statistically, a gender gap can be defined as the difference between the proportion of women and the proportion of men who support a particular politician, party, or policy position. In the 2020 presidential election, the winning candidate, Democrat Joe Biden, received 57 percent of women’s votes, compared with 45 percent of men’s, resulting in a gender gap of twelve percentage points.

The 2018 and 2020 congressional elections were filled with dramatic developments for women candidates. But in very different ways. The 2018 election set a record, with a total of 126 women elected to serve in the US House and Senate, far surpassing the 110 who served in the prior Congress. There has not been a single election year jump this large since the historic “year of the woman” elections in 1992. But in 2018, the success of women candidates was entirely on the Democratic side of the aisle. In fact, as these historic numbers were being achieved, the number of Republican women serving in the House fell to its lowest number in several decades.

When Kamala Harris took the stage on November 7, 2020, dressed in all white as a gesture to the suffragists who had worked so hard for this moment, she celebrated the breaking of “one of the most substantial barriers that exists in our country” through her election as the first woman, the first Black, and the first South Asian vice president of the United States. She reflected on the women who had paved the way to her victory and the message that her success would send to young people – that they should “dream with ambition, lead with conviction, and see yourself in a way that others might not see you.” She highlighted her own mother’s journey to America at age 19,

Bose–Chaudhuri–Hocquenghem (BCH) codes form a large class of cyclic codes for correcting multiple random errors. This class of codes was first discovered by Hocquenghem in 1959 [] and independently by Bose and Chaudhuri in 1960 []. The first algorithm for decoding binary BCH codes was devised by Peterson in 1960 [].