JPS6130819A - Error correction device - Google Patents

Abstract

PURPOSE:To obtain a root of an error location polynomial at two-word error by referencing a ROM table of a less memory quantity in the error correction of a duplicated error correction Reed Solomon code. CONSTITUTION:In an error correcting device provided with a means introducing sigma1, sigma2 at each code word from the error location polynomial, the coefficient sigma1 obtained at each code of two words in the error word number is obtained by (k) and (l) expressing two terms of alpha<k>+alpha<l> and a ROM table is provided, in which the (k) and (l) coincident with the coefficient sigma2 in alpha<(k+l)> are stored in an address corresponding to the coefficients sigma1 and sigma2. Then the alpha<k> and alpha<l> obtained from the ROM table in response to the said coefficients sigma1, sigma2 obtained at each code word including an errow word are taken as roots alpha<i> and alpha<j> of the error location polynomial on a GF(2<n>).

Description

【発明の詳細な説明】 （技術分野〕 不発明ａＰＣＭ符号の誤）訂正装置に関すｚ。[Detailed description of the invention] (Technical field〕 Regarding an uninvented aPCM code error correction device.

（従来技術） ２重誤り訂正リードソロモンコードによ２誤り訂正は、
誤りが、ブロック中のどのワードに生じてい乙かを示す
誤シ位置数と、誤りが生じたワードにおける誤シの大き
さを示す変数をシンドロームから求めえことによう行わ
ｔ′Ｌｚ０これらは周知の通シ次のステップによって行
われえ。(Prior art) Two error correction using a double error correction Reed-Solomon code is as follows:
The number of error positions indicating in which word in a block the error occurs and the variable indicating the size of the error in the word where the error occurred can be determined from the syndrome. The process is carried out by the following steps.

（Ｉ）シンドローム計算 （６）誤りワード数の算出 （ホ）誤シ位ｌＩ＋￥数を根に持つ誤り位置多項式の算
出（支）上記誤シ位置多項式の根の算出 Ｍ誤りの大きさ會表す変数の算出 （ロ）誤り訂正 上記ステップｔｖ＋ｖ符号長ｍ＋４、情報ワード数ｍの
２重機シ訂正す−ドンロモンコードについて説明す乙。(I) Syndrome calculation (6) Calculation of the number of error words (e) Calculation of the error locator polynomial whose root is the error position lI + ¥ number (support) Calculation of the root of the above error position polynomial M Expressing the magnitude of the error Calculation of variables (b) Error correction The above step tv + v code length m + 4, number of information words m, double machine correction - Don Romon code will be explained.

この様な符号語は ｍ＋３ Ｒ（ト）＝Ｗｒｎ９Ｘ＋モーーＸｍ＋２十−−−−〜−
−９Ｗ□Ｘ＋Ｗｏ　　−一一一一−−−−−−−−（１
）と表わされ、各ワードＷは、ｎビット構成のＣＦ（２
ｎ）上の各元αノである。伝送又は記録再生過程で符号
語の第１桁及び７桁のワードに誤りが生じたときシンド
ローム全基にその誤り位置数α１、αＪを２根とする［
ｌ！４シ位置多項式τに）が次のごとく得られる。This kind of code word is m+3 R(g)=Wrn9X+mo-Xm+20-------
-9W□X+Wo -1111---(1
), and each word W is represented by CF(2
n) Each element α is above. When an error occurs in the first and seventh digit words of a code word during transmission or recording/reproduction process, let the number of error positions α1 and αJ be two roots of the entire syndrome [
l! The 4-position polynomial τ) is obtained as follows.

６（ト）−１＋イ、Ｘ＋むｘ”　　　　　−−−−−−
−−−−−−（２）ちなみにシンドロームを で定義す４とき で与えられ２ことは周知の通りである。但しすべての演
算はＧＦ（２”）上で行われる。つま！り　（１１式で
示すＲ（ト）のワードが２語誤ったとき、その誤り位置
がどこであえかけδ（ｘ）＝θを解いて得られ２２根α
′、α！、求めればよい。しかえにイ（２）−〇を解く
方法としては次の２つが知られてい乙。その１つは根が
αｒｒｒ４−’〜α０のいずれかであるはずだから、そ
れらのうちδ（ｘ）−〇を満足す２α１、αｊのみを根
とすえ方法であえ。これは２（ｍ＋４）回の複雑な演算
を要し、符号語が到来する時間間隔内に処理すえことが
難しくな乙ことが多い。２番目の方法は上記の手順ｋ１
１ｆ；’、の種々の値に対してあらかじめ行っておき、
それら？ｌ−ＲＯＭテーブルにおさめ、δ（ｘ）＝０を
満足す４２つの係数ｄ、ヌ全アドレスとして２根全瞬時
に求め乙方法でろえ。6(g)-1+a,X+mux" ---------
--------(2) Incidentally, it is well known that the syndrome is defined as 4 given by 2. However, all calculations are performed on GF(2"). In other words, (when there are two incorrect words in R (g) shown in equation 11, where is the error position and δ(x) = θ? Solved and obtained 22 roots α
′、α! , all you have to do is ask. However, the following two methods are known to solve A(2)-〇. One of them is that the roots should be any one of αrrr4-' to α0, so among them, only 2α1 and αj that satisfy δ(x)-0 can be used as roots. This requires 2(m+4) complex operations and is often difficult to process within the time interval in which the codeword arrives. The second method is step k1 above.
Perform this in advance for various values of 1f;',
Those? Store it in the l-ROM table, calculate 42 coefficients d that satisfy δ(x) = 0, and calculate them instantly as all addresses of 2, using method B.

このテーブルは２ｎビツトのアドレスを有し１アドレス
に２ｎビツトのデータを持つ必要かめＺので１’２ｎ）
Ｘ２”ビットを要し巨大なメモリ量に達し実現しがたい
ことが多い。例えばｎ　＝　８のとき、その量は１３１
０７２バイトにも達する。This table has 2n-bit addresses, and each address has 2n-bit data (1'2n).
It requires X2" bits and requires a huge amount of memory, which is often difficult to implement. For example, when n = 8, the amount is 131
It reaches 072 bytes.

（目的） 本発明は上記ＲＯＭテーブルのサイズをはＺかに小さく
　＋、て誤り位置多項式の２根を短時間に求めうる誤り
訂正装ｆ’に提供すｚｏ （実施例） 以下不発明を具体的に説明すふ為ｎ　＝　８とし原始多
項式Ｘ”　＋Ｘ’　＋Ｘ″十Ｘ”　＋　１＝　０全満足
す為原始元αを有するＧＦ　（２”）上の符号語を例に
とる。イ（ト）＝０を満足す４２根もＧＦ（２“）上の
元であふからべき表現の元α０〜αｆ０のいずれかに該
当する。各元を多項式表現したべき表現−多項例えば誤
シ位置多項式　に）の一つの係数　、がα６８と得られ
た場合には　（２）＝０管満足す４２根αとαＪは ＝α６＠＝αイ＋αＪ の関係にあ４はずだから表１よりα６“の多項式表現（
α７＋α１＋α“＋１）を、例えば（α７）と（α４＋
α”＋１）、即ちα７とα１Ｇｇの如く１組に分け、こ
うして出来た種々の組のいずれかにα１、αｌは対応す
ｚｏどの組に該当すえか￥′：ｉ　に）のもう一つの係
数　８が ＝αｔ、αｊ＝αｉ＋／ の関係に６．にとから上述の各組のべき表現の積αｉ＋
／のうち　、と一致すｚものを選び出すことによりただ
ちにα１及びα／を決定できる。こうして上記係数　、
及び　、に対応すえ２根α８とα１．求め、各　、及び
　、に対応すえアドレスを有す４８０Ｍテーブルの各対
応位置に２根α“とα／全収納す４０ 上述の組数は、例えば多項式−ｊ９．現の項ｉｊ！、’
に４とすれば、表２のように求められ７とな４゜宍　　
　２ 一般に生じうえ組数は項数ｆ：Ｐとすえと２（１）−１
組となる。次に前記原始多項式に対す２元のべき表現と
多項式表現の対応表の全てにわたって生じ１組のデータ
量は３バイト（α１、αｊ、α“＋／）であるからＲＯ
Ｍテーブルに必要な総メモリ量は３０５７　ｘ　３　＝
　９１７１バイトとなり前述の例に比べ７チとごくわず
かのメモリ量ですみ、短時間で誤）位置多項式の２根を
求め２ことができ２゜根の求め方けざ、、／、？アドレ
スとするＲＯＭテーブル参照によ乙ことは勿論であふ。(Purpose) The present invention provides an error correction system f' that can obtain the second root of an error locator polynomial in a short time by reducing the size of the ROM table by Z. In order to explain this, we will set n = 8 and take as an example a code word on GF (2'') which has a primitive element α to satisfy the primitive polynomial X'' + X' + X'' + 1 = 0. The 42 roots that satisfy A(G) = 0 are also elements on GF(2") and correspond to any of the elements α0 to αf0 of the power representation. A power representation where each element is expressed as a polynomial - a polynomial, for example, an error If one coefficient of the position polynomial ) is obtained as α68, the 42 roots α and αJ that satisfy (2) = 0 tube should be 4, so from Table 1 α6 “Polynomial representation of (
α7+α1+α“+1), for example, (α7) and (α4+
α”+1), that is, divided into one set such as α7 and α1Gg, and α1 and αl correspond to any of the various sets created in this way. 8 is = αt, αj = αi+/ 6. From ni to the product αi+ of the power expressions of each set above.
α1 and α/ can be immediately determined by selecting z of / that matches . Thus, the above coefficient,
Corresponding to and , the two roots α8 and α1. Find and store the two roots α'' and α/all in each corresponding position of the 480M table with the corresponding address for each , and .
If it is 4, it is calculated as shown in Table 2 and becomes 7.
2 In general, the number of pairs that occur is the number of terms f:P and 2(1)-1
Become a group. Next, since the amount of data for one set is 3 bytes (α1, αj, α“+/) that occurs over all the correspondence tables between the binary power expression and the polynomial expression for the primitive polynomial, RO
The total amount of memory required for M table is 3057 x 3 =
It is 9171 bytes, which requires a very small amount of memory (7 chips) compared to the previous example, and it is possible to find the 2nd root of the position polynomial in a short time.How to find the 2° root? Of course, it is possible to refer to the ROM table as the address.

更に符号語長ｍ＋４は一般に２ｎ−１に比べ数分の１で
あえことが多いので一つの組の中で積成分を除き一４以
上の元金含む組はＲＯＭテーブルから除外して良いので
上記メモリ量は更に減少させゐことができＺｏ （効　果） 上記のように不発明によれば２重誤り訂正リードソロモ
ンコードの誤り訂正において２語誤り時の誤シ位置多項
式の根を少いメモリｔのＲＯＭテーブル全参照すること
によって短時間で求め４ことが出来る。Furthermore, since the code word length m+4 is generally a fraction of 2n-1, it is often the case that the code word length m+4 is a fraction of that of 2n-1, so in one set, except for the product component, sets containing 14 or more principals can be excluded from the ROM table, so the above memory (Effect) As mentioned above, according to the invention, in error correction of a double error correction Reed-Solomon code, the root of the error position polynomial at the time of two word errors can be stored in a small amount of memory t. This can be determined in a short time by referring to the entire ROM table.

Claims (1)

【特許請求の範囲】[Claims] ｎビットからなるデータワードｍ語と、同じくｎビット
からなるパリテイワード４語で１ブロックを構成し、ｍ
＋４≦２＾ｎ−１としたＧＦ（２＾ｎ）上の２重誤り訂
正リードソロモン符号について、シンドロームの公知の
組み合わせ演算により判定されるべき１ブロック中の誤
りワード数に応じて誤り位置数を根に持つ誤り位置多項
式δ（ｘ）＝１＋δ＿１ｘ＋δ＿２ｘ＾２から各符号語
毎にδ＿１及びδ＿２を導出する手段を備えた誤り訂正
装置において、誤りワード数が２語の符号語毎に求めた
前記多項式δ（ｘ）の係数δ＿１を、α＾ｋ＋α＾ｌの
如く２項で表現したｋ及びｌより求めたα＾ｋ＾＋＾ｌ
のうち、上記係数δ＿２と一致するα＾ｋ＋α＾ｌ又は
ｋ及びｌを、上記係数δ＿１及びδ＿２に対応するアド
レス上に収納したＲＯＭテーブルを有し、上記誤りワー
ドを含む符号語毎に求めた上記係数δ＿１及びδ＿２に
応じて上記ＲＯＭテーブルより求めたα＾ｋ及びα＾ｌ
をＧＦ（２＾ｎ）上の誤り位置多項式の根α＾ｉ及びα
＾ｊとすることを特徴とする誤り訂正装置。One block consists of m data words consisting of n bits and 4 parity words consisting of n bits, and m
For a double error correction Reed-Solomon code on GF(2^n) with +4≦2^n-1, the number of error positions is calculated according to the number of error words in one block to be determined by a known combination operation of syndromes. In an error correction apparatus equipped with a means for deriving δ_1 and δ_2 for each code word from an error locator polynomial δ(x)=1+δ_1x+δ_2x^2 having roots as The coefficient δ_1 of the polynomial δ(x) is calculated from k and l expressed in two terms such as α^k+α^l
Among them, α^k+α^l or k and l, which match the above coefficient δ_2, are stored in the addresses corresponding to the above coefficients δ_1 and δ_2 in a ROM table, and are calculated for each code word including the above error word. α^k and α^l obtained from the above ROM table according to the above coefficients δ_1 and δ_2
are the roots α^i and α of the error locator polynomial on GF(2^n)
An error correction device characterized by ^j.