Patent Application: US-99583309-A

Abstract:
the present discloses a method for detection and correction of errors , based on the proposition of multidimensional error correcting code , presenting the first example of implementation of mecc called bchmd , that employ the bch or bch algebraic in each dimension of the set of symbols in the encoder and decoder sides of the communication system , in the error correcting code stage . especially the described method claimed by the present invention embraces bits in different dimensions , which improves performance , speed and capacity in the ecc .

Description:
the contained examples has the single purpose of demonstrate and exemplify one of many possible implementations of the proposed multidimensional error correcting code , without limitations , since that any other similar implementations is inside of the scope of the present invention , methodology and process the coding or decoding method of present invention comprises the following steps : ( a ) attribution of a dimension for each element that compose a symbol , of a message , message composed by symbols , which symbols are composed at least 2 bits , so if a message is composed by 234 symbols , n will be 234 , this 180 are k , so 180 symbols compose the message codified by the system , so we obtain 2t = 54 symbols , where each symbol is composed by at least one bit ( in the case of a single bit composes my symbol , the known bch can be employed or the bch algebraic , despite for symbols of m ≧ 2 , where m is the number of bits that compose each symbol or the dimension of the galois field defined by the formulation gf ( 2 m ), that define the ring composed by symbols of m bits , without limit of maximum size to be used in the symbol representation ; ( b ) grouping of dimensions in same position of symbols in a new ( depart ) symbol ; ( c ) parallel processing of each dimension using a error correcting code ; specifically the content of the message is binary information , composed by a set of symbols of pre - defined sizes and each symbol aggregates a set of elements . the elements of a binary message are its bits and each symbol must contain at least two bits to a multidimensional error correcting code being applied . the current invention also contemplates a system for coding and decoding employing the multidimensional error correcting code , described in this document . such system can be employed in transmission and reception of data by , for example , optic fiber , wireless networks such as 3g , wimax , wifi , satellite communication also in computer networks , also in applications as broadcasting of digital radio and digital television . in mathematics , in abstract algebra and information theory , a linear code is an important type of block code used in error correction codes and schemes . linear codes allow for more efficient encoding and decoding algorithms than other codes ( cf . syndrome decoding ) [ 18 , 19 , 20 ]. linear codes are applied in methods of transmitting symbols ( e . g ., bits ) on a communications channel [ 21 , 18 , 19 , 22 ] so that , if errors occur in the communication , some errors can be detected by the recipient of a message block . the “ codes ” in the linear code are blocks of symbols which are encoded using more symbols than the original value to be sent . a linear code of length n transmits blocks containing n symbols . for example , the “( 7 , 4 )” hamming code is a binary linear code which represents 4 - bit values each using 7 - bit values . in this way , the recipient can detect errors as severe as 2 bits per block . as there are 16 distinct 4 - bit values expressed in binary , the size of the ( 7 , 4 ) hamming code is sixteen . a linear code of length n and rank k is a linear subspace c with dimension k of the vector space where is the finite field with q elements [ 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 ] . such a code with parameter q is called a q - ary code ( e . g ., when q = 5 , the code is a 5 - ary code ). if q = 2 or q = 3 , the code is described as a binary code , or a ternary code respectively . remark : we want to give the usual standard basis because each coordinate represents a “ bit ” which is transmitted across a “ noisy channel ” with some small probability of transmission error ( a binary symmetric channel ). if some other basis is used then this model cannot be used and the hamming metric ( defined next ) does not measure the number of errors in transmission , as we want it to . in telecommunication a convolutional [ 18 , 19 , 33 , 34 , 35 , 36 ] code is a type of error correcting code in which ( a ) each information symbol of m - bit length to be encoded is transformed into an n - bit symbol , where m / n is the code rate ( n ≧ m ) and ( b ) the transformation is a function of the last k information symbols , where k is the constraint length of the code . simple convolutional codes are now giving way to turbo codes ( a class of high - performance ), a new class of iterated short convolutional codes that closely approach the theoretical limits imposed by shannon &# 39 ; s [ 21 ] theorem with much less decoding complexity than the viterbi algorithm on the long convolutional codes that would be required for the same performance . based on this code rs , rm and bch are considered linear block codes , in the specific case of rs it is classified as a convolutional code [ 37 , 38 , 32 ]. the multidimensional error correcting code named bchmd , has the purpose of codifying symbols , symbols that are galois fields — gf ( 2 m )— for m ≧ 2 . the bchmd code considers that an error correcting code ( ecc ) works because of vectorial spaces condition , which ecc are based . that is , a encoder like bch or rs only works because they represents a vectorial space of n - tuples , in which each tuple is linearly independent from each other . extending this idea , the bchmd was constructed ( the prototype ) based on the algebraic bch , for coding and decoding using a algebraic bch in each dimension of the galois fields independently , each bit of the galois field is considered in a different dimension , and each dimension of the galois field will be considered as a dimension of the vectorial space of the symbols to be encoded . in other words , when encoding a set of symbols with rs , these symbols are tuples of bits . each symbol is composed generally by 8 bits . therefore , when rs encode , let &# 39 ; s assume as configuration n = 63 , k = 57 , this means that the rs code encode 57 symbols of input , generating an output of 63 symbols , being 6 symbols of parity , joined with the k symbols of data . therefore each symbol in true is a tuple of bits and the message is a tuple of symbols . so each symbol can be seen as a vectorial space of dimension m , since each symbol is a galois field of space gf ( 2 m ), generally also in our example , adopt m = 8 . based on publications that discuss the subject of linear systems and linear and abstract algebra , each symbol dimension is linearly independent as each symbol is linearly independent of each one . in this sense the encoding process is performed for each dimension of the symbols and each dimension is considered linearly independently . the bchmd example is applied to symbols of more than one bit to demonstrate the pertinence and viability of the multidimensional error correcting code . take as example the ring gf ( 2 8 ), which means 256 quadratic residues or residual fields , also galois fields where each fields is linearly independent of others . it is desired to construct a code , multidimensional ecc ( mecc ) example of implementation bchmd , align in parallel m codes of bch algebraic for example , to encode or even to decode the bit stream of each dimension in the vectorial space . we can exemplify the following set of symbols input of the bchmd , that are residues of the ring gf ( 2 ) as : α 1 = 00000000 ; α 2 = 00000010 ; α 3 = 00000100 ; α 4 = 00001000 ; α 5 = 00010000 ; α 6 = 00100000 ; α 7 = 01000000 ; α 8 = 10000000 . the mapping of the symbols dimensions for each algebraic bch is denoted in this manner for the symbols α 1 , α 4 and α 8 : in this manner , the mecc - bchmd is capable to obtain performance superior to the similar rs code using the same configuration of the rs . the bchmd for instance in its worst case achieve the same performance as the similar rs in its best case . this performance improvement related to the rs is only possible due to the fact that the bchmd is working over the dimensions of the fields in an independent manner . from the perspective of a hardware implementation the bchmd is achieving better performances because of the algebraic implementation of its code , not employing sequential or iterative algorithms or even worse a sequence of iterative algorithms like the case of rs implementing the berlekamp - massey or euclidean approaches . another important factor is implementing the mecc - bchmd in hardware or in software the performance of the code is always superior to a rs implementation . for example , to implement rs ( berlekamp - massey ), it will be necessary 57 clock cycles to generate the parity symbols , using symbols of 8 bits each . for the bchmd example no matter the size of the field the number of clock cycles will be always the same because each dimension is processed independently , and in parallel in a hardware implementation , and the time is fixed for each dimension . this means that is possible to use higher symbols gf ( 2 m ), through m with values higher than 8 . if needed a code with n , k with higher values , this won &# 39 ; t be a problem , because in hardware implementation or even in a software implementation the bch or algebraic bch can operate in parallel in each dimension . the increase in the symbol size or in the number of symbols , increase only in area or code increasing . not affecting necessarily the performance of the system . in a hardware implementation the encoding time never increased above 0 . 5 ( half ) clock cycle , the frequency being not the measurement unit , but the machine clock cycles . this is possible because of the matrix implementation of the bch algebraic in the hardware prototype using hdl . so the approach for implementing the mecc - bchmd is to explore all possible parallelism exploring the linear independence of the fields . in this manner the processing time is deterministic , also using the algebraic approach . besides , the mecc - bchmd is not limited as the rs into the t and 2t , parameters of correction capacity and detection of errors , despite the vectorial space dimensional size it will determine the linear combination of maximum capacity of detection and correction of errors , also it is possible to denote that , as mentioned before , this linear independence of the fields that give us the ability to achieve as the worst case scenario for the mecc - bchmd as the maximum capacity of a similar rs code . since the mecc - bchmd explore this dimensionally using a bch algebraic the minimal difference is for the bchmd ( 63 , 57 ): so dimensionally i can correct up to 2 errors at same time , since my symbol has 8 bits for example this means 8 times 2 , that is 16 the maximum capacity of my multidimensional error correcting code compared with the similar rs . the mecc - bchmd has as capacity t * m , the combination of the linear spaces of the code . this configurations t and 2t are applied not for the code , but for vectorial dimensions and not to the symbols or the set of symbols , this configuration affects only the dimensional capacity of mecc - bchmd because it is based on algebraic bch , any other multidimensional error correcting code ( mecc ) will be affect in different manners , regarding values of dimensional capacity , but the resulting behavior will be same for all mecc . the comparison between mecc - bchmd and rs for the same configuration ( 63 , 57 ) and the possible error patterns that each code can detect and correct are demonstrate as follow : in the equations of combinations , all the possible linear combinations of error patterns that the specific configurations is capable to identify and correct for each error correcting code comparing the capacities of rs and bchmd are presented . the bchmd capacity is the linear combination of all vectorial dimensions to define all the possible pattern errors of each . one positive factor about this implementation of multidimensional error correcting code is the fact that the bigger the galois field gf ( 2 m ) the value of m , the higher the capacity of detection and correction of this code . in fig5 , the columns represents the vectorial dimensions ( d 0 , . . . , d m ), in the positions that appear filled with ‘ x ’, they represents the presence of introduced errors in the encoded message , for demonstration we are adopting a code ( 63 , 57 ), whose rs code is capable of correct only 3 symbols , with erroneously bit in the symbol , while the mecc - bchmd is capable of detect and correct all the errors inserted in the m dimensions of the symbols , gf means the galois fields that compose the message and the elements p represents the parity symbols of the message . the mecc - bchmd and rs codes were implemented for comparison using the configuration ( 63 , 57 ). below is table 1 , demonstrating the results of the synthesis in hardware of both codes , implemented using a vhdl description : for testing purposes of the capacity between the codes , was used the following set of errors introduced in a message , gradually , which means : first test introduce the error in the symbol 50 , second in the symbols 50 and 49 and so on , plus of course the twelve errors independently : ( reg_msg_err ( 50 )& lt ;=“ 00000001 ”; reg_msg_err ( 49 )& lt ;=“ 00000010 ”; reg_msg_err ( 48 )& lt ;=“ 00000100 ”; reg_msg_err ( 47 )& lt ;=“ 00001000 ”; reg_msg_err ( 46 )& lt ;=“ 00010000 ”; reg_msg_err ( 45 )& lt ;=“ 00100000 ”; reg_msg_err ( 44 )& lt ;=“ 01000000 ”; reg_msg_err ( 43 )& lt ;=“ 10000000 ”; reg_msg_err ( 42 )& lt ;=“ 00000011 ”; reg_msg_err ( 41 )& lt ;=“ 00001100 ”; reg_msg_err ( 40 )& lt ;=“ 00110000 ”; reg_msg_err ( 39 )& lt ;=“ 11000000 ”; reg_msg_err ( 38 )& lt ;=“ 11111111 ”;). in the end a set of at least 144 tests were performed sequentially and gradually to measure the eccs capacity and differences . the first test was to insert an error in the less significative bit of symbol 50 , until we insert all the errors in the message and test a message with the maximum possible number of errors in all dimensions and test the capacity of both codes in same conditions . after perform the tests in the hardware implementation , the results are complied in table 2 . in this table the relations between the codes were presented , depicting the capacity of correction of each code for the same configuration , one following the traditional approach the rs and the other the bchmd a implementation of mecc approach . each column represents the number of symbols with m bits , containing errors in the message , each line represents the code for each specific number of erroneously bits by number of symbols for example line of 3 bits in column of 4 symbols means 4 symbols containing 3 erroneously bits in the message symbol . the presented data denote the obtained results of the implementations of both codes in hardware implementation . this data do not represent the limits of this process of technique of mecc , but they represent one example of possible implementation developed to demonstrate the potential of the mecc . the process can be applied for any known model of ecc . the mecc - bchmd gives the ability of new configurations of n and k , generating even larger possibilities of ecc , also still possible the inclusion of other techniques to improve the performance of the multidimensional error correcting code . it gives yet the possibility of implementing impossible ecc schemes adopting rs as base . 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