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In August 1908, a white mob attacked Springfield, Illinois’s Black community, destroying homes and businesses and lynching two Black men. National outrage over the violence helped spur the movement that created the NAACP the following year.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
In computer science and telecommunications, Hamming codes are a family of linear error-correcting codes. Hamming codes can detect one-bit and two-bit errors, or correct one-bit errors without detection of uncorrected errors. By contrast, the simple parity code cannot correct errors, and can detect only an odd number of bits in error. Hamming codes are perfect codes, that is, they achieve the highest possible rate for codes with their block length and minimum distance of three. Richard W. Hamming invented Hamming codes in 1950 as a way of automatically correcting errors introduced by punched card readers. In his original paper, Hamming elaborated his general idea, but specifically focused on the Hamming(7,4) code which adds three parity bits to four bits of data. In mathematical terms, Hamming codes are a class of binary linear codes. For each integer r ≥ 2 there is a code-word with block length n = 2r − 1 and message length k = 2r − r − 1. Hence the rate of Hamming codes is R = k / n = 1 − r / (2r − 1), which is the highest possible for codes with minimum distance of three (i.e., the minimal number of bit changes needed to go from any code word to any other code word is three) and block length 2r − 1. The parity-check matrix of a Hamming code is constructed by listing all columns of length r that are non-zero, which means that the dual code of the Hamming code is the shortened Hadamard code, also known as a Simplex code. The parity-check matrix has the property that any two columns are pairwise linearly independent. Due to the limited redundancy that Hamming codes add to the data, they can only detect and correct errors when the error rate is low. This is the case in computer memory (usually RAM), where bit errors are extremely rare and Hamming codes are widely used. Memory with this correction system is known as ECC memory. In this context, an extended Hamming code having one extra parity bit is often used. Extended Hamming codes achieve a Hamming distance of four, which allows the decoder to distinguish between when at most one one-bit error occurs and when any two-bit errors occur. In this sense, extended Hamming codes are single-error correcting and double-error detecting, abbreviated as SECDED.
In statistics and coding theory, a Hamming space is usually the set of all 2 N {\displaystyle 2^{N}} binary strings of length N, where different binary strings are considered to be adjacent when they differ only in one position. The total distance between any two binary strings is then the total number of positions at which the corresponding bits are different, called the Hamming distance. Hamming spaces are named after American mathematician Richard Hamming, who introduced the concept in 1950. They are used in the theory of coding signals and transmission. More generally, a Hamming space can be defined over any alphabet (set) Q as the set of words of a fixed length N with letters from Q. If Q is a finite field, then a Hamming space over Q is an N-dimensional vector space over Q. In the typical, binary case, the field is thus GF(2) (also denoted by Z2). In coding theory, if Q has q elements, then any subset C (usually assumed of cardinality at least two) of the N-dimensional Hamming space over Q is called a q-ary code of length N; the elements of C are called codewords. In the case where C is a linear subspace of its Hamming space, it is called a linear code. A typical example of linear code is the Hamming code. Codes defined via a Hamming space necessarily have the same length for every codeword, so they are called block codes when it is necessary to distinguish them from variable-length codes that are defined by unique factorization on a monoid. The Hamming distance endows a Hamming space with a metric, which is essential in defining basic notions of coding theory such as error detecting and error correcting codes. Hamming spaces over non-field alphabets have also been considered, especially over finite rings (most notably over Z4) giving rise to modules instead of vector spaces and ring-linear codes (identified with submodules) instead of linear codes. The typical metric used in this case the Lee distance. There exist a Gray isometry between Z 2 2 m {\displaystyle \mathbb {Z} _{2}^{2m}} (i.e. GF(22m)) with the Hamming distance and Z 4 m {\displaystyle \mathbb {Z} _{4}^{m}} (also denoted as GR(4,m)) with the Lee distance.
In coding theory, Hamming(7,4) is a linear error-correcting code that encodes four bits of data into seven bits by adding three parity bits. It is a member of a larger family of Hamming codes, but the term Hamming code often refers to this specific code that Richard W. Hamming introduced in 1950. At the time, Hamming worked at Bell Telephone Laboratories and was frustrated with the error-prone punched card reader, which is why he started working on error-correcting codes. The Hamming code adds three additional check bits to every four data bits of the message. Hamming's (7,4) algorithm can correct any single-bit error, or detect all single-bit and two-bit errors. In other words, the minimal Hamming distance between any two correct codewords is 3, and received words can be correctly decoded if they are at a distance of at most one from the codeword that was transmitted by the sender. This means that for transmission medium situations where burst errors do not occur, Hamming's (7,4) code is effective (as the medium would have to be extremely noisy for two out of seven bits to be flipped). In quantum information, the Hamming (7,4) is used as the base for the Steane code, a type of CSS code used for quantum error correction.
Richard Wesley Hamming (February 11, 1915 – January 7, 1998) was an American mathematician whose work had many implications for computer engineering and telecommunications. His contributions include the Hamming code (which makes use of a Hamming matrix), the Hamming window, Hamming numbers, the sphere-packing or Hamming bound, Hamming graph concepts, and the Hamming distance. Born in Chicago, Hamming attended the University of Chicago, the University of Nebraska and the University of Illinois Urbana-Champaign, where he wrote his doctoral thesis in mathematics under the supervision of Waldemar Trjitzinsky (1901–1973). In April 1945, he joined the Manhattan Project at the Los Alamos Laboratory, where he programmed the IBM calculating machines that computed the solution to equations provided by the project's physicists. He left to join the Bell Telephone Laboratories in 1946. Over the next fifteen years, he was involved in nearly all of the laboratories' most prominent achievements. For his work, he received the ACM Turing Award in 1968, being its third recipient. After retiring from Bell Labs in 1976, Hamming took a position at the Naval Postgraduate School in Monterey, California, where he worked as an adjunct professor and senior lecturer in computer science, and devoted himself to teaching and writing books. He delivered his last lecture in December 1997, just a few weeks before he died from a heart attack on January 7, 1998.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Mae Jemison, aboard Space Shuttle Endeavour in 1992.